Questions number 19 and 20 are Assertion and Reason based questions. Two statements are given, one labelled Assertion (A) and the other labelled Reason (R). Select the correct answer from the codes (A), (B), (C) and (D) as given below.
(A) Both Assertion (A) and Reason (R) are true and Reason (R) is...
CBSE CLASS 12 • MATHEMATICS • CHAPTER 10
Vectors
316 distinct questions across 483 board paper set appearances (2022–2026).
The position vectors of points P and Q are $\vec{p}$ and $\vec{q}$ respectively. The point R divides line segment PQ in the ratio $3:1$ and S is the mid-point of line segment PR. The position vector of S is : (A) $\frac{\vec{p} + 3\vec{q}}{4}$ (B) $\frac{\vec{p} + 3\vec{q}}{8}$ (C) $\frac{5\vec{p} +...
The Cartesian equation of the line passing through the point $(1, -3, 2)$ and parallel to the line: $\vec{r} = $2 + \lambda$\hat{i} + \lambda\hat{j} + $2\lambda - 1$\hat{k}$ is (A) $\frac{x - 1}{2} = \frac{y + 3}{0} = \frac{z - 2}{-1}$ (B) $\frac{x + 1}{1} = \frac{y - 3}{1} = \frac{z + 2}{2}$ (C) $\...
Assertion (A) : The vectors
$\vec{a} = 6\hat{i} + 2\hat{j} - 8\hat{k}$
$\vec{b} = 10\hat{i} - 2\hat{j} - 6\hat{k}$
$\vec{c} = 4\hat{i} - 4\hat{j} + 2\hat{k}$
represent the sides of a right angled triangle.
Reason (R) : Three non-zero vectors of which none of two are collinear forms a triangle if th...
The value of m for which the points with position vectors $-\hat{i} - \hat{j} + 2\hat{k}$, $2\hat{i} + m\hat{j} + 5\hat{k}$ and $3\hat{i} + 11\hat{j} + 6\hat{k}$ are collinear, is
(A) $8$
(B) $-8$
(C) $2$
(D) $\\frac{5}{2}$
Assertion (A) : Lines given by $x = py + q$, $z = ry + s$ and $x = p'y + q'$, $z = r'y + s'$ are perpendicular to each other when $pp' + rr' = 1$.
Reason (R) : Two lines $\vec{r} = \vec{a}_1 + \lambda \vec{b}_1$ and $\vec{r} = \vec{a}_2 + \mu \vec{b}_2$ are perpendicular to each other if $\vec{b}_1 ...
Vectors $\vec{a} = 3\hat{i} - 2\hat{j} + 2\hat{k}$ and $\vec{b} = \hat{i} + 2\hat{k}$ represent the two adjacent sides of a parallelogram. Find the vectors representing its diagonals and hence find their lengths.
If $| \vec{a} | = 5$ and $-2 \le \lambda \le 1$, then the sum of greatest and the smallest value of $| \lambda \vec{a} |$ is
(A) $-5$
(B) $5$
(C) $10$
(D) $15$
If for two unit vectors $\vec{a}$ and $\vec{b}$, $|\vec{a} + 2\vec{b}| = |2\vec{a} - \vec{b}|$, then find the angle between $\vec{a}$ and $\vec{b}$.
If $$3\hat{i} - 2\hat{j} + 5\hat{k}$ \times $4\hat{i} + p\hat{j} + q\hat{k}$ = \vec{0}$, then the values of $p$ and $q$ are :
(A) $p = -\frac{2}{3}, q = \frac{5}{3}$
(B) $p = -\frac{8}{3}, q = \frac{20}{3}$
(C) $p = \frac{20}{3}, q = -\frac{8}{3}$
(D) $p = 0, q = 0$
(a) Let two rods placed on the ground be represented by vectors $4\hat{i} - \hat{j} + 3\hat{k}$ and $-2\hat{i} + \hat{j} - 2\hat{k}$. Find a vector representing a flag-post of height 5 m that has to be erected perpendicular to both the rods. OR (b) A unit vector $\vec{a}$ is such that it makes an an...
(a) Find a point on the line $\frac{x - 2}{3} = \frac{1 - y}{2} = \frac{z - 3}{2}$ at a distance of $\sqrt{2}$ units from the point (1, 2, 3).
OR
(b) Find the shortest distance between the lines
$\vec{r} = $4 + \lambda$\hat{i} + $2\lambda - 1$\hat{j} - 3\lambda\hat{k}$
$\vec{r} = $1 + 2\mu$\hat{i} +...
If vectors $\vec{a} = 3\hat{i} - 2\hat{j} + 4\hat{k}$ and $\vec{b} = \lambda\hat{i} + \hat{j}$ are perpendicular to each other, then the value of $\lambda$ is :
(A) $3$
(B) $-3$
(C) $\frac{2}{3}$
(D) $-\frac{2}{3}$
(a) Find the equation of a line passing through the point $(1, -1, 2)$ and parallel to x-axis. Also, find the shortest distance between this line and the line $\vec{r} = 2\hat{i} + \hat{j} - \hat{k} + \mu$3\hat{i} - 5\hat{j} + 2\hat{k}$$्ले.
OR
(b) Find the equations of the lines AC and BD, if coord...
(a) Let $\vec{\alpha}, \vec{\beta}, \vec{\gamma}$ be three non-zero vectors, such that $\vec{\alpha} \times \vec{\beta} = \vec{\gamma}$ and $\vec{\beta} \times \vec{\gamma} = \vec{\alpha}$. Show that $\alpha, \beta, \gamma$ are mutually perpendicular vectors such that $|\vec{\beta}| = 1$ and $|\vec{...
If $|\vec{a}| = 1$, $|\vec{b}| = 2$ and $\vec{a} \cdot \vec{b} = 2$, then the value of $|\vec{a} + \vec{b}|$ is :
(A) 9
(B) 3
(C) $-3$
(D) 2
(a) Find the shortest distance between the lines $l_1$ and $l_2$ given by :
$l_1 : \vec{r} = \hat{i} + 2\hat{j} - 4\hat{k} + \lambda $4\hat{i} + 6\hat{j} + 12\hat{k}$$
and $l_2 : \vec{r} = 3\hat{i} + 3\hat{j} - 5\hat{k} + \mu $6\hat{i} + 9\hat{j} + 18\hat{k}$$
OR
(b) Show that the lines $\frac{x - 1...
Let $\theta$ be the angle between two unit vectors $\hat{a}$ and $\hat{b}$ such that $\sin \theta = \frac{3}{5}$.
Then, $\hat{a} \cdot \hat{b}$ is equal to :
(A) $\pm \frac{3}{5}$
(B) $\pm \frac{3}{4}$
(C) $\pm \frac{4}{5}$
(D) $\pm \frac{4}{3}$
If $\vec{a} + \vec{b} + \vec{c} = \vec{0}$, $|\vec{a}| = \sqrt{37}$, $|\vec{b}| = 3$ and $|\vec{c}| = 4$, then angle between $\vec{b}$ and $\vec{c}$ is
(A) $\frac{\pi}{6}$
(B) $\frac{\pi}{4}$
(C) $\frac{\pi}{3}$
(D) $\frac{\pi}{2}$
(a) Verify that lines given by $\vec{r} = $1 - \lambda$\hat{i} + (\lambda - 2)\hat{j} + $3 - 2\lambda$\hat{k}$ and $\vec{r} = (\mu + 1)\hat{i} + $2\mu - 1$\hat{j} - $2\mu + 1$\hat{k}$ are skew lines. Hence, find shortest distance between the lines.
OR
(b) During a cricket match, the position of th...
The diagonals of a parallelogram are given by $\vec{a} = 2\hat{i} - \hat{j} + \hat{k}$ and $\vec{b} = \hat{i} + 3\hat{j} - \hat{k}$. Find the area of the parallelogram.
(a) Two friends while flying kites from different locations, find the strings of their kites crossing each other. The strings can be represented by vectors $\vec{a} = 3\hat{i} + \hat{j} + 2\hat{k}$ and $\vec{b} = 2\hat{i} - 2\hat{j} + 4\hat{k}$. Determine the angle formed between the kite strings. A...
(a) Show that the area of a parallelogram whose diagonals are represented by $\vec{a}$ and $\vec{b}$ is given by $\frac{1}{2} |\vec{a} \times \vec{b}|$. Also find the area of a parallelogram whose diagonals are $2\hat{i} - \hat{j} + \hat{k}$ and $\hat{i} + 3\hat{j} - \hat{k}$.
OR
(b) Find the equati...
Let $|\vec{a}| = 5$ and $-2 \le \lambda \le 1$. Then, the range of $|\lambda \vec{a}|$ is :
(A) $[5, 10]$
(B) $[-2, 5]$
(C) $[-2, 1]$
(D) $[-10, 5]$
(a) If $\vec{a}$ and $\vec{b}$ are position vectors of point A and point B respectively, find the position vector of point C on BA produced such that $\text{BC} = 3\text{BA}$.
OR
(b) Vector $\vec{r}$ is inclined at equal angles to the three axes x, y and z. If magnitude of $\vec{r}$ is $5\sqrt{3}$...
A student tries to tie ropes, parallel to each other from one end of the wall to the other. If one rope is along the vector $3 \hat{i} + 15 \hat{j} + 6 \hat{k}$ and the other is along the vector $2 \hat{i} + 10 \hat{j} + \lambda \hat{k}$, then the value of $\lambda$ is :
(A) $6$
(B) $1$
(C) $\fra...
If $|\vec{a} + \vec{b}| = |\vec{a} - \vec{b}|$ for any two vectors, then vectors $\vec{a}$ and $\vec{b}$ are :
(A) orthogonal vectors
(B) parallel to each other
(C) unit vectors
(D) collinear vectors
(a) The scalar product of the vector $\vec{a} = \hat{i} - \hat{j} + 2\hat{k}$ with a unit vector along sum of vectors $\vec{b} = 2\hat{i} - 4\hat{j} + 5\hat{k}$ and $\vec{c} = \lambda\hat{i} - 2\hat{j} - 3\hat{k}$ is equal to $1$. Find the value of $\lambda$.
OR
(b) Find the shortest distance betw...
Three friends A, B and C move out from the same location O at the same time in three different directions to reach their destinations. They move out on straight paths and decide that A and B after reaching their destinations will meet up with C at his predecided destination, following straight paths...
The unit vector perpendicular to both vectors $\hat{i} + \hat{k}$ and $\hat{i} - \hat{k}$ is : (A) 2$\hat{j}$ (B) $\hat{j}$ (C) $\frac{\hat{i} - \hat{k}}{\sqrt{2}}$ (D) $\frac{\hat{i} + \hat{k}}{\sqrt{2}}$
If $\vec{a} = 2\hat{i} - \hat{j} + \hat{k}$ and $\vec{b} = \hat{i} + \hat{j} - \hat{k}$, then $\vec{a}$ and $\vec{b}$ are :
(A) collinear vectors which are not parallel
(B) parallel vectors
(C) perpendicular vectors
(D) unit vectors
If $| \vec{a} | = 2$ and $-3 \le k \le 2$, then $| k \vec{a} | \in$ :
(A) $[-6, 4]$
(B) $[0, 4]$
(C) $[4, 6]$
(D) $[0, 6]$
Find the position vector of point C which divides the line segment joining points A and B having position vectors $\hat{i} + 2\hat{j} - \hat{k}$ and $-\hat{i} + \hat{j} + \hat{k}$ respectively in the ratio $4 : 1$ externally. Further, find $|\vec{AB}| : |\vec{BC}|$.
If $\vec{a}$ and $\vec{b}$ are two vectors such that $| \vec{a} | = 1, | \vec{b} | = 2$ and $\vec{a} \cdot \vec{b} = \sqrt{3}$, then the angle between $2\vec{a}$ and $-\vec{b}$ is :
(A) $\frac{\pi}{6}$
(B) $\frac{\pi}{3}$
(C) $\frac{5\pi}{6}$
(D) $\frac{11\pi}{6}$
The vectors $\vec{a} = 2\hat{i} - \hat{j} + \hat{k}$, $\vec{b} = \hat{i} - 3\hat{j} - 5\hat{k}$ and $\vec{c} = -3\hat{i} + 4\hat{j} + 4\hat{k}$ represents the sides of
(A) an equilateral triangle
(B) an obtuse-angled triangle
(C) an isosceles triangle
(D) a right-angled triangle
Let $\vec{a}$ be any vector such that $|\vec{a}| = a$. The value of
$|\vec{a} \times \hat{i}|^2 + |\vec{a} \times \hat{j}|^2 + |\vec{a} \times \hat{k}|^2$ is :
(A) $a^2$
(B) $2a^2$
(C) $3a^2$
(D) 0
An instructor at the astronomical centre shows three among the brightest stars in a particular constellation. Assume that the telescope is located at O(0, 0, 0) and the three stars have their locations at the points D, A and V having position vectors $2\hat{i} + 3\hat{j} + 4\hat{k}$, $7\hat{i} + 5\h...
OR
(b) Find the shortest distance between the lines $L_1$ & $L_2$ given below :
$L_1 :$ The line passing through $(2, -1, 1)$ and parallel to $\frac{x}{1} = \frac{y}{1} = \frac{z}{3}$
$L_2 : \vec{r} = \hat{i} + $2\mu + 1$\hat{j} - (\mu + 2)\hat{k}$.
(a) Find the co-ordinates of the foot of the perpendicular drawn from the point $(2, 3, -8)$ to the line $\frac{4-x}{2} = \frac{y}{6} = \frac{1-z}{3}$. Also, find the perpendicular distance of the given point from the line. OR (b) Find the shortest distance between the lines $L_1$ & $L_2$ given belo...
(a) Find the shortest distance between the lines $\frac{x - 8}{3} = \frac{y + 9}{-16} = \frac{z - 10}{7}$ and $\frac{x - 15}{3} = \frac{y - 29}{8} = \frac{z - 5}{-5}$. OR (b) Find the point of intersection of the lines $\vec{r} = \hat{i} - \hat{j} + 6\hat{k} + \lambda$3\hat{i} - \hat{k}$$, and $\vec...
The value of p for which the vectors $2\hat{i} + p\hat{j} + \hat{k}$ and $-4\hat{i} - 6\hat{j} + 26\hat{k}$ are perpendicular to each other, is : (a) 3 (b) -3 (c) $-\frac{17}{3}$ (d) $\frac{17}{3}$
Unit vector along $\vec{PQ}$, where coordinates of P and Q respectively are $(2, 1, -1)$ and $(4, 4, -7)$, is
(A) $2\hat{i} + 3\hat{j} - 6\hat{k}$
(B) $-2\hat{i} - 3\hat{j} + 6\hat{k}$
(C) $\frac{-2\hat{i}}{7} - \frac{3\hat{j}}{7} + \frac{6\hat{k}}{7}$
(D) $\frac{2\hat{i}}{7} + \frac{3\hat{j}}{7} - ...
A unit vector along the vector $4\hat{i} - 3\hat{k}$ is :
(a) $\frac{1}{7}$4\hat{i} - 3\hat{k}$$
(b) $\frac{1}{5}$4\hat{i} - 3\hat{k}$$
(c) $\frac{1}{\sqrt{7}}$4\hat{i} - 3\hat{k}$$
(d) $\frac{1}{\sqrt{5}}$4\hat{i} - 3\hat{k}$$
If $\vec{a}$, $\vec{b}$ and $$\vec{a} + \vec{b}$$ are all unit vectors and $\theta$ is the angle between $\vec{a}$ and $\vec{b}$, then the value of $\theta$ is :
(a) $\frac{2\pi}{3}$
(b) $\frac{5\pi}{6}$
(c) $\frac{\pi}{3}$
(d) $\frac{\pi}{6}$
(a) If $\vec{a}$, $\vec{b}$ and $\vec{c}$ are three vectors such that $| \vec{a} | = 7$, $| \vec{b} | = 24$, $| \vec{c} | = 25$ and $\vec{a} + \vec{b} + \vec{c} = \vec{0}$, then find the value of $\vec{a} \cdot \vec{b} + \vec{b} \cdot \vec{c} + \vec{c} \cdot \vec{a}$.
OR
(b) If a line makes angles $...
(a) If $\vec{a}$, $\vec{b}$, $\vec{c}$ and $\vec{d}$ are four non-zero vectors such that $\vec{a} \times \vec{b} = \vec{c} \times \vec{d}$ and $\vec{a} \times \vec{c} = 4\vec{b} \times \vec{d}$, then show that $$\vec{a} - 2\vec{d}$$ is parallel to $$2\vec{b} - \vec{c}$$ where $\vec{a} \neq 2\vec{d}$...
$\vec{a}$ and $\vec{b}$ are two unit vectors such that $|2\vec{a} + 3\vec{b}| = |3\vec{a} - 2\vec{b}|$. Find the angle between $\vec{a}$ and $\vec{b}$.
Write the projection of the vector $$\vec{b} + \vec{c}$$ on the vector $\vec{a}$, where $\vec{a} = 2\hat{i} - 2\hat{j} + \hat{k}$, $\vec{b} = \hat{i} + 2\hat{j} - 2\hat{k}$ and $\vec{c} = 2\hat{i} - \hat{j} + 4\hat{k}$.
Find the distance of the point $(1, -2, 9)$ from the point of intersection of the line $\vec{r} = 4\hat{i} + 2\hat{j} + 7\hat{k} + \lambda$3\hat{i} + 4\hat{j} + 2\hat{k}$$ and the plane $\vec{r} \cdot $\hat{i} - \hat{j} + \hat{k}$ = 10$.
Find the equations of the planes passing through the line of intersection of the planes $\vec{r} \cdot $\hat{i} + 3\hat{j}$ = 6$ and $\vec{r} \cdot $3\hat{i} - \hat{j} - 4\hat{k}$ = 0$, which are at a distance of 1 unit from the origin.
If $\vec{a} = \hat{i} + \hat{j} - 2\hat{k}$, $\vec{b} = -\hat{i} + 2\hat{j} + 2\hat{k}$ and $\vec{c} = -\hat{i} + 2\hat{j} - \hat{k}$ are three vectors, then find a vector perpendicular to both the vectors $$\vec{a} + \vec{b}$$ and $$\vec{b} - \vec{c}$$.
If $\vec{a}$, $\vec{b}$ and $\vec{c}$ are unit vectors such that $\vec{a} + \vec{b} + \vec{c} = \vec{0}$, then find the value of $\vec{a} \cdot \vec{b} + \vec{b} \cdot \vec{c} + \vec{c} \cdot \vec{a}$.
Find the equation of the plane passing through the intersection of the planes $\vec{r} \cdot $2\hat{i} + 2\hat{j} - 3\hat{k}$ = 7$ and $\vec{r} \cdot $2\hat{i} + 5\hat{j} + 3\hat{k}$ = 9$ and through the point $(2, 1, 3)$.
The value of $p$ for which vectors $\hat{i} + 2\hat{j} + 3\hat{k}$ and $2\hat{i} - p\hat{j} + \hat{k}$ are perpendicular to each other is
(A) $0$
(B) $1$
(C) $\frac{5}{2}$
(D) $-\frac{5}{2}$
Vector of magnitude $3$ making equal angles with $x$ and $y$ axes and perpendicular to $z$ axis is
(A) $\hat{i} + 2\sqrt{2} \hat{j}$
(B) $3\hat{k}$
(C) $\frac{3\sqrt{2}}{2} \hat{i} + \frac{3\sqrt{2}}{2} \hat{j}$
(D) $\sqrt{3} \hat{i} + \sqrt{3} \hat{j} + \sqrt{3} \hat{k}$
If $\overrightarrow{\text{AB}} = \hat{\jmath} + \hat{k}$ and $\overrightarrow{\text{AC}} = 3\hat{\imath} - \hat{\jmath} + 4\hat{k}$ represent the two vectors along the sides $\text{AB}$ and $\text{AC}$ of $\Delta\text{ABC}$, prove that the median $\overrightarrow{\text{AD}} = \frac{\overrightarrow{\...
A line passing through the points $\text{A}(1, 2, 3)$ and $\text{B}(5, 8, 11)$ intersects the line $\vec{r} = 4\hat{i} + \hat{j} + \lambda$5\hat{i} + 2\hat{j} + \hat{k}$$. Find the co-ordinates of the point of intersection. Hence, write the equation of a line passing through the point of intersectio...
For two vectors $\vec{a}$ and $\vec{b}$
Assertion (A) : $|\vec{a} imes \vec{b}|^2 + $\vec{a} \cdot \vec{b}$^2 = |\vec{a}|^2 |\vec{b}|^2$
Reason (R) : $|\vec{a} imes \vec{b}| = $\vec{a} \cdot \vec{b}$ an heta, \left$ heta
eq \frac{\pi}{2}
ight$$
For any two vectors $\vec{a}$ and $\vec{b}$, which of the following statements is always true ?
(A) $\vec{a} \cdot \vec{b} \le |\vec{a}|\,|\vec{b}|$
(B) $|\vec{a} + \vec{b}| \ge |\vec{a}| + |\vec{b}|$
(C) $|\vec{a} - \vec{b}| = |\vec{a}| - |\vec{b}|$
(D) $|\vec{a} \times \vec{b}| \ge |\vec{a}|\,|\ve...
For any two vectors $\vec{a}$ and $\vec{b}$, which of the following statements is always true ?
(A) $\vec{a} \cdot \vec{b} \le |\vec{a}| |\vec{b}|$
(B) $|\vec{a} + \vec{b}| \ge |\vec{a}| + |\vec{b}|$
(C) $|\vec{a} - \vec{b}| = |\vec{a}| - |\vec{b}|$
(D) $|\vec{a} \times \vec{b}| \ge |\vec{a}| |\vec{...
If $$\vec{a} + \vec{b}$ \cdot $\vec{a} - \vec{b}$ = 198$ and $|\vec{a}| = 10 |\vec{b}|$, then :
(A) $|\vec{a}| = \sqrt{2}$
(B) $|\vec{b}| = \sqrt{2}$
(C) $|\vec{b}| = 10\sqrt{2}$
(D) $|\vec{a}| = \frac{10}{\sqrt{2}}$
If vectors $\vec{a} = 3\hat{i} + 2\hat{j} + \lambda\hat{k}$ and $\vec{b} = 2\hat{i} - 4\hat{j} + 5\hat{k}$, represent the two strips of the Red Cross sign placed outside a doctor's clinic, then the value of $\lambda$ is : (A) 1 (B) $\frac{5}{2}$ (C) $\frac{2}{5}$ (D) 0
(a) Three honey bees were found flying along the vectors $\vec{a} = 2\hat{i} - 3\hat{j} + \hat{k}$, $\vec{b} = 4\hat{j} - 2\hat{k}$ and $\vec{c} = 3\hat{i} + 2\hat{k}$ respectively. Find the value of $\lambda$ such that the path for $\vec{a} + \lambda \vec{b}$ is perpendicular to $\vec{c}$. OR (b) I...
Let three toys A, B and C be placed in the same straight line. If the position vectors of A, B and C are $5\hat{i} - 2\hat{j}$, $5\hat{i} + 8\hat{j}$ and $a\hat{i} - 52\hat{j}$ respectively, find the value of 'a'.
If $\vec{a}$, $\vec{b}$ and $\vec{c}$ are unit vectors, then prove that $|\vec{a}-\vec{b}|^2 + |\vec{b}-\vec{c}|^2 + |\vec{c}-\vec{a}|^2 \leq 9$.
(a) Find the foot of the perpendicular from the point $(0, 2, 3)$ on the line $\frac{-x - 3}{-5} = \frac{1 - y}{-2} = \frac{3z + 12}{9}$ and hence find the length of the perpendicular. OR (b) Find the value of p if the shortest distance between the lines $\vec{r} = $\hat{i} + 2\hat{j} + \hat{k}$ + \...
(a) Find the value of $\lambda$, if the points $(-1, -1, 2)$, $(2, 8, \lambda)$ and $(3, 11, 6)$ are collinear.
OR
(b) $\vec{a}$ and $\vec{b}$ are two co-initial vectors forming the adjacent sides of a parallelogram such that $|\vec{a}|=10$, $|\vec{b}|=2$ and $\vec{a}\cdot\vec{b}=12$. Find the are...
Find the angle between the lines
$\vec{r}=$3+2\lambda$\hat{i}-$2-2\lambda$\hat{j}+$6+2\lambda$\hat{k}$ and
$\vec{r}=$2\hat{j}-5\hat{k}$+\mu$6\hat{i}+3\hat{j}+2\hat{k}$$.
If $\hat{a}$, $\hat{b}$ and $\hat{c}$ are unit vectors such that $\hat{a} \cdot \hat{b} = \hat{a} \cdot \hat{c} = 0$ and the angle between $\hat{b}$ and $\hat{c}$ is $\frac{\pi}{6}$, then prove that $\hat{a} = \pm 2 \left$\hat{b} \times \hat{c}\right$$.
If $\vec{a}$ and $\vec{b}$ are two non-zero vectors such that $$\vec{a} + \vec{b}$ \perp \vec{a}$ and $$2\vec{a} + \vec{b}$ \perp \vec{b}$, then prove that $|\vec{b}| = \sqrt{2} |\vec{a}|$.
The equation of a line parallel to the vector $3\hat{i} + \hat{j} + 2\hat{k}$ and passing through the point $(4, -3, 7)$ is :
(A) $x = 4t + 3$, $y = -3t + 1$, $z = 7t + 2$
(B) $x = 3t + 4$, $y = t + 3$, $z = 2t + 7$
(C) $x = 3t + 4$, $y = t - 3$, $z = 2t + 7$
(D) $x = 3t + 4$, $y = -t + 3$, $z = 2t ...
(a) Find the shortest distance between the lines: $\frac{x+1}{2} = \frac{y-1}{1} = \frac{z-9}{-3}$ and $\frac{x-3}{2} = \frac{y+15}{-7} = \frac{z-9}{5}$.
OR
(b) Find the image $A'$ of the point $A(2, 1, 2)$ in the line $l: \vec{r} = 4\hat{i} + 2\hat{j} + 2\hat{k} + \lambda $\hat{i} - \hat{j} - \hat{...
If the sides AB and AC of $\Delta$ ABC are represented by vectors $\hat{j} + \hat{k}$ and 3$\hat{i} - \hat{j} + 4\hat{k}$ respectively, then the length of the median through A on BC is : (A) 2$\sqrt{2}$ units (B) $\sqrt{18}$ units (C) $\frac{\sqrt{34}}{2}$ units (D) $\frac{\sqrt{48}}{2}$ units
Let $\vec{a}$ be a position vector whose tip is the point $(2, -3)$. If $\overrightarrow{\text{AB}} = \vec{a}$, where coordinates of $\text{A}$ are $(-4, 5)$, then the coordinates of $\text{B}$ are :
(A) $(-2, -2)$
(B) $(2, -2)$
(C) $(-2, 2)$
(D) $(2, 2)$
(a) Find a vector of magnitude $5$ which is perpendicular to both the vectors $3\hat{i} - 2\hat{j} + \hat{k}$ and $4\hat{i} + 3\hat{j} - 2\hat{k}$.
OR
(b) Let $\vec{a}$, $\vec{b}$ and $\vec{c}$ be three vectors such that $\vec{a}\cdot\vec{b} = \vec{a}\cdot\vec{c}$ and $\vec{a}\times\vec{b} = \vec{...
(a) If $\vec{a} + \vec{b} + \vec{c} = \vec{0}$ such that $|\vec{a}| = 3$, $|\vec{b}| = 5$, $|\vec{c}| = 7$, then find the angle between $\vec{a}$ and $\vec{b}$.
OR
(b) If $\vec{a}$ and $\vec{b}$ are unit vectors inclined with each other at an angle $\theta$, then prove that $\frac{1}{2} |\vec{a} -...
Assertion (A) : If $|\vec{a} imes \vec{b}|^2 + |\vec{a} \cdot \vec{b}|^2 = 256$ and $|\vec{b}| = 8$, then $|\vec{a}| = 2$.
Reason (R) : $\sin^2 heta + \cos^2 heta = 1$ and $|\vec{a} imes \vec{b}| = |\vec{a}| |\vec{b}| \sin heta$ and $\vec{a} \cdot \vec{b} = |\vec{a}| |\vec{b}| \cos heta$.
(a) Find the distance of the point $P(2, 4, -1)$ from the line $\frac{x + 5}{1} = \frac{y + 3}{4} = \frac{z - 6}{-9}$.
OR
(b) Let the position vectors of the points A, B and C be $3\hat{i} - \hat{j} - 2\hat{k}$, $\hat{i} + 2\hat{j} - \hat{k}$ and $\hat{i} + 5\hat{j} + 3\hat{k}$ respectively. Find ...
For any two vectors $\vec{a}$ and $\vec{b}$, which of the following statements is always true ?
(A) $\vec{a} \cdot \vec{b} \ge |\vec{a}| |\vec{b}|$
(B) $\vec{a} \cdot \vec{b} = |\vec{a}| |\vec{b}|$
(C) $\vec{a} \cdot \vec{b} \le |\vec{a}| |\vec{b}|$
(D) $\vec{a} \cdot \vec{b} < |\vec{a}| |\vec{b}|$
Assertion (A) : For two non-zero vectors $\vec{a}$ and $\vec{b}$, $\vec{a} \cdot \vec{b} = \vec{b} \cdot \vec{a}$. Reason (R) : For two non-zero vectors $\vec{a}$ and $\vec{b}$, $\vec{a} \times \vec{b} = \vec{b} \times \vec{a}$.
The position vectors of vertices of $\Delta ABC$ are $A$2\hat{i} - \hat{j} + \hat{k}$$, $B$\hat{i} - 3\hat{j} - 5\hat{k}$$ and $C$3\hat{i} - 4\hat{j} - 4\hat{k}$$. Find all the angles of $\Delta ABC$.
Assertion (A) : Projection of $\vec{a}$ on $\vec{b}$ is same as projection of $\vec{b}$ on $\vec{a}$. Reason (R) : Angle between $\vec{a}$ and $\vec{b}$ is same as angle between $\vec{b}$ and $\vec{a}$ numerically.
Find a vector of magnitude 4 units perpendicular to each of the vectors $2\hat{i} - \hat{j} + \hat{k}$ and $\hat{i} + \hat{j} - \hat{k}$ and hence verify your answer.
Given $\vec{a} = 2\hat{i} - \hat{j} + \hat{k}$, $\vec{b} = 3\hat{i} - \hat{k}$ and $\vec{c} = 2\hat{i} + \hat{j} - 2\hat{k}$. Find a vector $\vec{d}$ which is perpendicular to both $\vec{a}$ and $\vec{b}$ and $\vec{c} \cdot \vec{d} = 3$.
Find the projection of vector $$\vec{b} + \vec{c}$$ on vector $\vec{a}$, where $\vec{a} = 2\hat{i} + 2\hat{j} + \hat{k}$, $\vec{b} = \hat{i} + 3\hat{j} + \hat{k}$ and $\vec{c} = \hat{i} + \hat{k}$.
What is the value of $\frac{\text{projection of } \vec{a} \text{ on } \vec{b}}{\text{projection of } \vec{b} \text{ on } \vec{a}}$ for vectors $\vec{a}=2\hat{i}-3\hat{j}-6\hat{k}$ and $\vec{b}=2\hat{i}-2\hat{j}+\hat{k}$?
If $\vec{a}$ and $\vec{b}$ are two vectors such that $\vec{a} \cdot \vec{b} > 0$ and $|\vec{a} \cdot \vec{b}| = |\vec{a} \times \vec{b}|$, then the angle between $\vec{a}$ and $\vec{b}$ is:
Find all the vectors of magnitude $3\sqrt{3}$ which are collinear to vector $\hat{i} + \hat{j} + \hat{k}$.
If $\vec{a} + \vec{b} = \hat{i}$ and $\vec{a} = 2\hat{i} - 2\hat{j} + 2\hat{k}$, then $|\vec{b}|$ equals :
(a) $\sqrt{14}$
(b) 3
(c) $\sqrt{12}$
(d) $\sqrt{17}$
Position vectors of the points A, B and C as shown in the figure below are $\vec{a}$, $\vec{b}$ and $\vec{c}$ respectively. If $\vec{AC} = \frac{5}{4} \vec{AB}$, express $\vec{c}$ in terms of $\vec{a}$ and $\vec{b}$.
If $\vec{a}, \vec{b}, \vec{c}$ are three non-zero unequal vectors such that $\vec{a} \cdot \vec{b} = \vec{a} \cdot \vec{c}$, then find the angle between $\vec{a}$ and $\vec{b} - \vec{c}$.
In $\Delta \text{ABC}$, $\vec{\text{AB}} = \hat{i} + \hat{j} + 2\hat{k}$ and $\vec{\text{AC}} = 3\hat{i} - \hat{j} + 4\hat{k}$. If D is mid-point of BC, then vector $\vec{\text{AD}}$ is equal to :
(a) $4\hat{i} + 6\hat{k}$
(b) $2\hat{i} - 2\hat{j} + 2\hat{k}$
(c) $\hat{i} - \hat{j} + \hat{k}$
(d...
$\vec{a}$ and $\vec{b}$ are two non-zero vectors such that the projection of $\vec{a}$ on $\vec{b}$ is 0. The angle between $\vec{a}$ and $\vec{b}$ is :
(a) $\frac{\pi}{2}$
(b) $\pi$
(c) $\frac{\pi}{4}$
(d) 0
Assertion (A) : A line through the points $(4, 7, 8)$ and $(2, 3, 4)$ is parallel to a line through the points $(-1, -2, 1)$ and $(1, 2, 5)$.
Reason (R): Lines $\vec{r} = \vec{a_1} + \lambda \vec{b_1}$ and $\vec{r} = \vec{a_2} + \mu \vec{b_2}$ are parallel if $\vec{b_1} \cdot \vec{b_2} = 0$.
If $\theta$ is the angle between two vectors $\vec{a}$ and $\vec{b}$, then $\vec{a} \cdot \vec{b} \geq 0$ only when :
(a) $0 < \theta < \frac{\pi}{2}$
(b) $0 \leq \theta \leq \frac{\pi}{2}$
(c) $0 < \theta < \pi$
(d) $0 \leq \theta \leq \pi$
Assertion (A) : The lines $\vec{r} = \vec{a_1} + \lambda\vec{b_1}$ and $\vec{r} = \vec{a_2} + \mu\vec{b_2}$ are perpendicular, when $\vec{b_1} \cdot \vec{b_2} = 0$.
Reason (R) : The angle $\theta$ between the lines $\vec{r} = \vec{a_1} + \lambda\vec{b_1}$ and $\vec{r} = \vec{a_2} + \mu\vec{b_2}$ is ...
If the projection of the vector $\hat{i} + \hat{j} + \hat{k}$ on the vector $p\hat{i} + \hat{j} - 2\hat{k}$ is $\frac{1}{3}$, then find the value(s) of $p$.
Two vectors $\vec{a} = a_1 \hat{i} + a_2 \hat{j} + a_3 \hat{k}$ and $\vec{b} = b_1 \hat{i} + b_2 \hat{j} + b_3 \hat{k}$ are collinear if
(a) $a_1 b_1 + a_2 b_2 + a_3 b_3 = 0$
(b) $\frac{a_1}{b_1} = \frac{a_2}{b_2} = \frac{a_3}{b_3}$
(c) $a_1 = b_1, a_2 = b_2, a_3 = b_3$
(d) $a_1 + a_2 + a_3 = b_1 + ...
(a) If the vectors $\vec{a}$ and $\vec{b}$ are such that $|\vec{a}| = 3$, $|\vec{b}| = \frac{2}{3}$ and $\vec{a} \times \vec{b}$ is a unit vector, then find the angle between $\vec{a}$ and $\vec{b}$. OR (b) Find the area of a parallelogram whose adjacent sides are determined by the vectors $\vec{a} ...
(a) Find the coordinates of the foot of the perpendicular drawn from the point $P(0, 2, 3)$ to the line $\frac{x+3}{5} = \frac{y-1}{2} = \frac{z+4}{3}$.
OR
(b) Three vectors $\vec{a}$, $\vec{b}$ and $\vec{c}$ satisfy the condition $\vec{a} + \vec{b} + \vec{c} = \vec{0}$. Evaluate the quantity $\mu...
Find the distance between the lines :
$\vec{r} = $\hat{i} + 2\hat{j} - 4\hat{k}$ + \lambda$2\hat{i} + 3\hat{j} + 6\hat{k}$$;
$\vec{r} = $3\hat{i} + 3\hat{j} - 5\hat{k}$ + \mu$4\hat{i} + 6\hat{j} + 12\hat{k}$$
The projection of vector $\hat{i}$ on the vector $\hat{i} + \hat{j} + 2\hat{k}$ is :
(a) $\frac{1}{\sqrt{6}}$
(b) $\sqrt{6}$
(c) $\frac{2}{\sqrt{6}}$
(d) $\frac{3}{\sqrt{6}}$
If ABCD is a parallelogram and AC and BD are its diagonals, then $\vec{AC} + \vec{BD}$ is :
(a) $2\vec{DA}$
(b) $2\vec{AB}$
(c) $2\vec{BC}$
(d) $2\vec{BD}$
In a parallelogram PQRS, $\vec{PQ} = 3\hat{i} - 2\hat{j} + 2\hat{k}$ and $\vec{PS} = -\hat{i} - 2\hat{k}$. Find $|\vec{PR}|$ and $|\vec{QS}|.$
The two adjacent sides of a parallelogram are represented by vectors $2\hat{i} - 4\hat{j} + 5\hat{k}$ and $\hat{i} - 2\hat{j} - 3\hat{k}$. Find the unit vector parallel to one of its diagonals. Also, find the area of the parallelogram. OR If $\vec{a} = 2\hat{i} + 2\hat{j} + 3\hat{k}$, $\vec{b} = -\h...
(a) If $|\vec{a} imes \vec{b}|^2 + |\vec{a} \cdot \vec{b}|^2 = 400$ and $|\vec{b}| = 5$, then find the value of $|\vec{a}|$.
OR
(b) Find all the possible vectors of magnitude $5\sqrt{3}$ which are equally inclined to the coordinate axes.
The value of p for which vectors $\hat{i} + 2\hat{j} + 3\hat{k}$ and $2\hat{i} - p\hat{j} + \hat{k}$ are perpendicular to each other is
(A) $0$
(B) $1$
(C) $\\frac{5}{2}$
(D) $-\\frac{5}{2}$
If $|\vec{a}| = 8$, $|\vec{b}| = 3$ and $|\vec{a} \ imes \vec{b}| = 12$, then the value of $|\vec{a} \cdot \vec{b}|$
(A) $6\sqrt{3}$
(B) $8\sqrt{3}$
(C) $12\sqrt{3}$
(D) $3\sqrt{12}$
If $|\vec{a}| = 8$, $|\vec{b}| = 3$ and $|\vec{a} \times \vec{b}| = 12$, then the value of $|\vec{a} \cdot \vec{b}|$
(A) $6\sqrt{3}$
(B) $8\sqrt{3}$
(C) $12\sqrt{3}$
(D) $3\sqrt{12}$
A vector $\vec{a}$ of magnitude $14$ has direction ratios $<2, 3, -6>$. Find the projection of the vector $\vec{a}$ on $\hat{i}$.
If $|\vec{a}| = 8$, $|\vec{b}| = 3$ and $|\vec{a} imes \vec{b}| = 12$, then the value of $|\vec{a} \cdot \vec{b}|$
(A) $6\sqrt{3}$
(B) $8\sqrt{3}$
(C) $12\sqrt{3}$
(D) $3\sqrt{12}$
For two vectors $\vec{a}$ and $\vec{b}$
Assertion (A) : $|\vec{a} \times \vec{b}|^2 + $\vec{a} \cdot \vec{b}$^2 = |\vec{a}|^2 |\vec{b}|^2$
Reason (R) : $|\vec{a} \times \vec{b}| = $\vec{a} \cdot \vec{b}$ \tan \theta$, $\left$\theta \neq \frac{\pi}{2}\right$$
Find the co-ordinates of the point on the line $\vec{r} = -\hat{\jmath} + 3\hat{k} + \lambda$2\hat{\imath} - 2\hat{\jmath} + \hat{k}$$ such that the sum of co-ordinates is $3$.
Vector of magnitude $3$ making equal angles with $x$ and $y$ axes and perpendicular to $z$ axis is
(A) $\hat{i} + 2\sqrt{2}\hat{j}$
(B) $3\hat{k}$
(C) $\frac{3\sqrt{2}}{2}\hat{i} + \frac{3\sqrt{2}}{2}\hat{j}$
(D) $\sqrt{3}\hat{i} + \sqrt{3}\hat{j} + \sqrt{3}\hat{k}$
If $\vec{\text{AB}} = \hat{\text{j}} + \hat{\text{k}}$ and $\vec{\text{AC}} = 3\hat{\text{i}} - \hat{\text{j}} + 4\hat{\text{k}}$ represent the two vectors along the sides $\text{AB}$ and $\text{AC}$ of $\Delta \text{ABC}$, prove that the median $\vec{\text{AD}} = \frac{\vec{\text{AB}} + \vec{\text{...
A line passing through the points $ ext{A}(1, 2, 3)$ and $ ext{B}(5, 8, 11)$ intersects the line $\vec{r} = 4\hat{i} + \hat{j} + \lambda$5\hat{i} + 2\hat{j} + \hat{k}$$. Find the co-ordinates of the point of intersection. Hence, write the equation of a line passing through the point of intersection ...
Let $\vec{a}$ and $\vec{b}$ be two unit vectors and $\theta$ is the angle between them. Then $\vec{a} + 2\vec{b}$ is a unit vector if :
(A) $\theta = \frac{\pi}{2}$
(B) $\theta = \frac{2\pi}{3}$
(C) $\theta = \frac{3\pi}{4}$
(D) $\theta = \pi$
The area (in square units) of a parallelogram, whose diagonals are $\vec{a} = \hat{i} + \hat{j} - 2\hat{k}$ and $\vec{b} = \hat{i} - 3\hat{j} + 4\hat{k}$ is :
(A) $2\sqrt{14}$
(B) $\sqrt{14}$
(C) $\sqrt{38}$
(D) $2\sqrt{6}$
24. Find a unit vector perpendicular to both of the vectors $\overrightarrow{AB}$ and $\overrightarrow{BC}$, where A, B and C are $(3, -1, 2)$, $(1, -1, -3)$ and $(4, -3, 1)$, respectively.
25. Find the image of the point $(2, -1, 5)$ in the line $\vec{r} = $11\hat{i} - 2\hat{j} - 8\hat{k}$ + \lambda$10\hat{i} - 4\hat{j} - 11\hat{k}$$.
If $\vec{a}$, $\vec{b}$ and $\vec{c}$ are mutually perpendicular vectors of equal magnitudes, show that the vector $\vec{a} + \vec{b} + \vec{c}$ is equally inclined to $\vec{a}$, $\vec{b}$ and $\vec{c}$.
Find the shortest distance between the lines :
$\vec{r} = $\hat{i} + 2\hat{j} + 3\hat{k}$ + \lambda$\hat{i} - 3\hat{j} + 2\hat{k}$$, and
$\vec{r} = $4\hat{i} + 5\hat{j} + 6\hat{k}$ + \mu$2\hat{i} + 3\hat{j} + \hat{k}$$.
If $$\vec{a} + \vec{b}$ \cdot $\vec{a} - \vec{b}$ = 198$ and $|\vec{a}| = 10\,|\vec{b}|$, then :
(A) $|\vec{a}| = \sqrt{2}$
(B) $|\vec{b}| = \sqrt{2}$
(C) $|\vec{b}| = 10\sqrt{2}$
(D) $|\vec{a}| = \frac{10}{\sqrt{2}}$
For any two unit vectors $\vec{a}$ and $\vec{b}$, if $|\vec{a} + 2\vec{b}| = |2\vec{a} - \vec{b}|$, then find the angle between $\vec{a}$ and $\vec{b}$.
Find the vector equation of the line passing through the origin and perpendicular to both the lines $\vec{r} = 2\hat{i} - \hat{j} + 2\hat{k} + \lambda$3\hat{i} + 4\hat{j} + 2\hat{k}$$ and $\vec{r} = \mu$\hat{i} - \hat{j} + \hat{k}$$.
(a) If the lines $\frac{x - 3}{1} = \frac{1 - y}{1} = \frac{z + 2}{p}$ and $\frac{2 - x}{3} = \frac{y + 1}{5} = \frac{z + 56}{2p}$ are perpendicular to each other, then find the value(s) of $p$.
OR
(b) Find the vector equation of a line passing through the origin and perpendicular to both the line...
(b) The opposite sides of a square are along the lines:
$\vec{r} = \hat{i} + 2\hat{j} - 4\hat{k} + \lambda$2\hat{i} + 3\hat{j} + 6\hat{k}$$
$\vec{r} = 3\hat{i} + 3\hat{j} - 5\hat{k} + \mu$2\hat{i} + 3\hat{j} + 6\hat{k}$$
If the direction ratios of the other pair of opposite sides are $\langle -3, 6,...
(a) Represent the equations of lines $l_1$ and $l_2$ in vector form and check whether they are intersecting or not.
$$l_1: \frac{x+3}{-3} = \frac{y-1}{1} = \frac{z-5}{5}$$
$$l_2: \frac{x+1}{-1} = \frac{2-y}{-2} = \frac{z-5}{5}$$
OR
(b) Opposite sides of a square are along the lines :
$$\vec{r} = \ha...
Assertion (A) : The vectors $\vec{a}$ and $$-2\vec{a}$$, where $\vec{a} \neq \vec{0}$ are collinear vectors.\nReason (R) : $\vec{a} \cdot $-2\vec{a}$ = 0$.
(b) Find the vector equation of a line passing through the origin and perpendicular to both the lines $\vec{r} = 2\hat{i} - \hat{j} + 2\hat{k} + \lambda$3\hat{i} + 4\hat{j} + 2\hat{k}$$ and $\vec{r} = \mu$\hat{i} - \hat{j} + \hat{k}$$.
Opposite sides of a square are along the lines :
$\vec{r} = \hat{i} + 2\hat{j} - 4\hat{k} + \lambda$2\hat{i} + 3\hat{j} + 6\hat{k}$$
$\vec{r} = 3\hat{i} + 3\hat{j} - 5\hat{k} + \mu$2\hat{i} + 3\hat{j} + 6\hat{k}$$
Find the area of the square if direction ratios of other pair of opposite sides of the...
Assertion (A) : Vectors $\vec{a}$ and $$-2\vec{a}$$, where $\vec{a} \neq \vec{0}$, are collinear vectors.
Reason (R) : $\vec{a} \cdot $-2\vec{a}$ = 0$.
Assertion (A) : The vectors $\vec{a}$ and $$-2\vec{a}$$, where $\vec{a} \neq \vec{0}$ are collinear vectors.
Reason (R) : $\vec{a} \cdot $-2\vec{a}$ = 0$.
OR Find the vector equation of a line passing through the origin and perpendicular to both the lines $\vec{r} = 2\hat{i} - \hat{j} + 2\hat{k} + \lambda$3\hat{i} + 4\hat{j} + 2\hat{k}$$ and $\vec{r} = \mu$\hat{i} - \hat{j} + \hat{k}$$.
OR
Opposite sides of a square are along the lines :
$\vec{r} = \hat{i} + 2\hat{j} - 4\hat{k} + \lambda$2\hat{i} + 3\hat{j} + 6\hat{k}$$
$\vec{r} = 3\hat{i} + 3\hat{j} - 5\hat{k} + \mu$2\hat{i} + 3\hat{j} + 6\hat{k}$$
Find the area of the square if direction ratios of other pair of opposite sides of ...
If the position vectors of three points A, B and C are $3\hat{i} + \hat{j}$, $5\hat{i} + 6\hat{j} - 3\hat{k}$ and $4\hat{j}$ respectively, then show that they form an isosceles triangle.
Find the equation of a line (in vector and cartesian form) that passes through the point of intersection of lines $\frac{x - 1}{2} = \frac{y - 2}{3} = \frac{z - 3}{4}$ and $\frac{x - 4}{5} = \frac{y - 1}{2} = z$ and is parallel to the vector $3\hat{i} + 2\hat{j} - 8\hat{k}$.
If $$2\hat{i} + 3\hat{j} + \hat{k}$$ and $$2\hat{i} + \hat{j} - \hat{k}$$ represent the sides $\vec{AB}$ and $\vec{AC}$ respectively of $\Delta ABC$, find the vector representing the median through A.
Find the vector and cartesian equations of the line passing through the point of intersection of the lines $\vec{r} = $\hat{i} + \hat{j} - \hat{k}$ + \lambda$3\hat{i} - \hat{j}$$ and $\vec{r} = $4\hat{i} - \hat{k}$ + \mu$2\hat{i} + 3\hat{k}$$ and parallel to the line $\frac{x - 1}{-2} = \frac{7 - y}...
OR (b) A unit vector $\vec{a}$ is such that it makes an angle $\frac{\pi}{4}$ with x-axis, $\frac{\pi}{3}$ with y-axis and an acute angle $\theta$ with z-axis. Find $\theta$ and the components of $\vec{a}$.
If in a parallelogram PQRS, $\vec{PQ} = 2\hat{i} + 3\hat{j} + \hat{k}$ and $\vec{QR} = \hat{i} - 2\hat{j} + \hat{k}$, then find the unit vectors parallel to the diagonals $\vec{PR}$ and $\vec{SQ}$.
If position vector $\vec{p}$ of a point $(24, n)$ is such that $|\vec{p}| = 25$, then the value of $n$ is : (A) $\pm 49$ (B) $\pm 5$ (C) $\pm 1$ (D) $\pm 7$
If position vector $\vec{\text{p}}$ of a point (24, n) is such that $| \vec{\text{p}} | = 25$, then the value of n is :
(A) $\pm 49$
(B) $\pm 5$
(C) $\pm 1$
(D) $\pm 7$
(a) Three honey bees were found flying along the vectors $\vec{\text{a}} = 2\hat{\text{i}} - 3\hat{\text{j}} + \hat{\text{k}}$, $\vec{\text{b}} = 4\hat{\text{j}} - 2\hat{\text{k}}$ and $\vec{\text{c}} = 3\hat{\text{i}} + 2\hat{\text{k}}$ respectively. Find the value of $\lambda$ such that the path f...
(b) Find the value of p if the shortest distance between the lines $\vec{r} = $\hat{i} + 2\hat{j} + \hat{k}$ + \lambda$\hat{i} - \hat{j} + \hat{k}$$ and $\vec{r} = $p\hat{i} - \hat{j} - \hat{k}$ + \mu$2\hat{i} + \hat{j} + 2\hat{k}$$ is $\frac{3}{\sqrt{2}}$ units.
If vectors $\vec{\text{a}} = 3\hat{\text{i}} + 2\hat{\text{j}} + \lambda\hat{\text{k}}$ and $\vec{\text{b}} = 2\hat{\text{i}} - 4\hat{\text{j}} + 5\hat{\text{k}}$, represent the two strips of the Red Cross sign placed outside a doctor's clinic, then the value of $\lambda$ is :
(A) $1$
(B) $\frac{5}{...
If position vector $\vec{\text{p}}$ of a point $(24, \text{n})$ is such that $| \vec{\text{p}} | = 25$, then the value of $\text{n}$ is :
(A) $\pm 49$
(B) $\pm 5$
(C) $\pm 1$
(D) $\pm 7$
(a) Let three toys A, B and C be placed in the same straight line. If the position vectors of A, B and C are $55\hat{i} - 2\hat{j}$, $5\hat{i} + 8\hat{j}$ and $a\hat{i} - 52\hat{j}$ respectively, find the value of 'a'. OR (b) If $\vec{a}$, $\vec{b}$ and $\vec{c}$ are unit vectors, then prove that $|...
If $A(3, 5, 1)$ and $B(1, 1, 5)$ are the end points of the vector $\vec{\text{AB}}$, then the direction cosines of $\vec{\text{AB}}$ are :
(A) $-2, -4, 4$
(B) $-\frac{1}{3}, -\frac{2}{3}, \frac{2}{3}$
(C) $-\frac{1}{2}, -1, 1$
(D) $\frac{1}{3}, \frac{2}{3}, \frac{2}{3}$
The unit vector perpendicular to both the vectors $\vec{a} = $-\hat{i} + \hat{j} + \hat{k}$$ and $\vec{b} = $-\hat{i} - \hat{j} - \hat{k}$$ is :
(A) $-2\hat{j} + 2\hat{k}$
(B) $2\hat{j} - 2\hat{k}$
(C) $-\frac{1}{2}\hat{j} + \frac{1}{2}\hat{k}$
(D) $-\frac{1}{\sqrt{2}}\hat{j} + \frac{1}{\sqrt{2}...
If the line $\frac{1 - x}{-3} = \frac{2y - 2}{8} = \frac{z + 6}{5}$ is parallel to the line $\vec{r} = $2\hat{i} - 3\hat{j} + 5\hat{k}$ + \lambda$-6\hat{i} + \lambda\hat{j} - 10\hat{k}$$, then the value of $\lambda$ is :
(A) $8$
(B) $-8$
(C) $4$
(D) $-4$
The projection of $\vec{a} = $2\hat{i} - \hat{j} + \hat{k}$$ on $\vec{b} = $\hat{i} - 2\hat{j} + \hat{k}$$ is :
(A) $\frac{3}{\sqrt{6}}$
(B) $-\frac{5}{\sqrt{6}}$
(C) $\frac{2}{\sqrt{6}}$
(D) $\frac{5}{\sqrt{6}}$
(a) If $\vec{a} + \vec{b} + \vec{c} = \vec{0}$ and $|\vec{a}| = 3$, $|\vec{b}| = 5$ and $|\vec{c}| = 7$, then find the angle between $\vec{a}$ and $\vec{b}$.
OR
(b) Find the area of a parallelogram whose adjacent sides are given by the vectors $\vec{a} = \hat{i} - 2\hat{j} + 3\hat{k}$ and $\vec{b}...
Find the vector equation of a line, passing through the point with position vector $\hat{i} - 2\hat{j} - 3\hat{k}$ and parallel to the line joining the points with position vectors $\hat{i} - \hat{j} + 4\hat{k}$ and $2\hat{i} + \hat{j} + 2\hat{k}$. Also, write its cartesian equation.
A bird flies through a distance in a straight line given by the vector $\hat{i} + 2\hat{j} + \hat{k}$. A man standing beside a straight metro rail track given by $\vec{r} = $3+\lambda$\hat{i} + $2\lambda-1$\hat{j} + 3\lambda\hat{k}$ is observing the bird. The projected length of its flight on the me...
The distance of the point with position vector $3\hat{i} + 4\hat{j} + 5\hat{k}$ from the y-axis is
(A) $4$ units
(B) $\sqrt{34}$ units
(C) $5$ units
(D) $5\sqrt{2}$ units
If $\vec{a} = 3\hat{i} + 2\hat{j} + 4\hat{k}$, $\vec{b} = \hat{i} + \hat{j} - 3\hat{k}$ and $\vec{c} = 6\hat{i} - \hat{j} + 2\hat{k}$ are three given vectors, then $$2\vec{a}.\hat{i}$\hat{i} - (\vec{b}.\hat{j})\hat{j} + (\vec{c}.\hat{k})\hat{k}$ is same as the vector
(A) $\vec{a}$
(B) $\vec{b} + \ve...
Assertion (A): Given two non-zero vectors $\vec{a}$ and $\vec{b}$ . If $\vec{r}$ is another non-zero vector such that $\vec{r} \times $\vec{a} + \vec{b}$ = \vec{0}$. Then $\vec{r}$ is perpendicular to $\vec{a} \times \vec{b}$ .
Reason (R): The vector $$\vec{a} + \vec{b}$$ is perpendicular to the pla...
The two vectors $\hat{i} + \hat{j} + \hat{k}$ and $3\hat{i} - \hat{j} + 3\hat{k}$ represent the two sides $OA$ and $OB$, respectively of a $\Delta OAB$, where $O$ is the origin. The point $P$ lies on $AB$ such that $OP$ is a median. Find the area of the parallelogram formed by the two adjacent sides...
Find the distance of the point $(2, -1, 3)$ from the line
$\vec{r} = $2\hat{i} - \hat{j} + 2\hat{k}$ + \mu$3\hat{i} + 6\hat{j} + 2\hat{k}$$
measured parallel to the $z$-axis.
Find the point of intersection of the line $\vec{r} = $3\hat{i} + \hat{k}$ + \mu$\hat{i} + \hat{j} + \hat{k}$$ and the line through $(2, -1, 1)$ parallel to the $z$-axis. How far is this point from the $z$-axis?
The value of $x$ for which $x$\hat{i} + \hat{j} + \hat{k}$$ is a unit vector, is :
(A) $1$
(B) $\pm\sqrt{3}$
(C) $\pm\frac{1}{\sqrt{3}}$
(D) $\pm 3$
(a) If $\vec{a}$, $\vec{b}$ and $\vec{c}$ are three vectors such that $\vec{a} + \vec{b} + \vec{c} = \vec{0}$ and $|\vec{a}| = 2$, $|\vec{b}| = 3$ and $|\vec{c}| = 4$, then find the value of $\vec{a} \cdot \vec{b} + \vec{b} \cdot \vec{c} + \vec{c} \cdot \vec{a}$.
(b) Let $\vec{a} = 2\hat{i} + 3\hat{j} + 4\hat{k}$, $\vec{b} = 4\hat{i} - 3\hat{j} + 9\hat{k}$ and $\vec{c} = 3\hat{i} - 2\hat{j} + 6\hat{k}$. Find a vector $\vec{d}$ which is perpendicular to both $\vec{a}$ and $\vec{b}$ and $\vec{c} \cdot \vec{d} = 39$.
(a) Write the nature of the lines $\frac{x-1}{4} = \frac{y-2}{6} = \frac{z-3}{8}$ and $\frac{x-2}{2} = \frac{y-4}{3} = \frac{z-5}{4}$. Also, find the shortest distance between them.
OR
(b) Show that the lines
$\vec{r} = $2\hat{j} - 3\hat{k}$ + \lambda$\hat{i} + 2\hat{j} + 3\hat{k}$$ and
$\vec{r} =...
Two vectors $\vec{a}$ and $\vec{b}$ are such that $|\vec{a} \times \vec{b}| = \vec{a} \cdot \vec{b}$. The angle between the two vectors is :
(A) $30^{\circ}$
(B) $60^{\circ}$
(C) $45^{\circ}$
(D) $90^{\circ}$
Two vectors $\vec{a}$ and $\vec{b}$ are such that $|\vec{a} \times \vec{b}| = \vec{a} . \vec{b}$. The angle between the two vectors is :
(A) $30^\circ$
(B) $60^\circ$
(C) $45^\circ$
(D) $90^\circ$
If $\hat{a}$, $\hat{b}$ and $\hat{c}$ are unit vectors such that $\hat{a}.\hat{b} = \hat{a}.\hat{c} = 0$ and the angle between $\hat{b}$ and $\hat{c}$ is $\frac{\pi}{6}$, then prove that $\hat{a} = \pm 2 $\hat{b} \times \hat{c}$$.
Two vectors $\vec{\text{a}}$ and $\vec{\text{b}}$ are such that $|\vec{\text{a}} \times \vec{\text{b}}| = \vec{\text{a}} \cdot \vec{\text{b}}$. The angle between the two vectors is :
(A) $30^{\circ}$
(B) $60^{\circ}$
(C) $45^{\circ}$
(D) $90^{\circ}$
(a) Find the value of $\lambda$, if the points $(-1, -1, 2)$, $(2, 8, \lambda)$ and $(3, 11, 6)$ are collinear.\nOR\n(b) $\vec{a}$ and $\vec{b}$ are two co-initial vectors forming the adjacent sides of a parallelogram such that $|\vec{a}| = 10$, $|\vec{b}| = 2$ and $\vec{a} \cdot \vec{b} = 12$. Find...
The vector with terminal point $\text{A}(2, -3, 5)$ and initial point $\text{B}(3, -4, 7)$ is :
(A) $\hat{i} - \hat{j} + 2\hat{k}$
(B) $\hat{i} + \hat{j} + 2\hat{k}$
(C) $-\hat{i} - \hat{j} - 2\hat{k}$
(D) $-\hat{i} + \hat{j} - 2\hat{k}$
In the given figure, $ABCD$ is a parallelogram. If $\overrightarrow{AB} = 2\hat{i} - 4\hat{j} + 5\hat{k}$ and $\overrightarrow{DB} = 3\hat{i} - 6\hat{j} + 2\hat{k}$, then find $\overrightarrow{AD}$ and hence find the area of parallelogram $ABCD$.
If vector $\vec{a} = 3\hat{i} + 2\hat{j} - \hat{k}$ and vector $\vec{b} = \hat{i} - \hat{j} + \hat{k}$, then which of the following is correct ?
(A) $\vec{a} \parallel \vec{b}$
(B) $\vec{a} \perp \vec{b}$
(C) $|\vec{b}| > |\vec{a}|$
(D) $|\vec{a}| = |\vec{b}|$
If $\vec{\alpha} = \hat{i} - 4\hat{j} + 9\hat{k}$ and $\vec{\beta} = 2\hat{i} - 8\hat{j} + \lambda\hat{k}$ are two mutually parallel vectors, then $\lambda$ is equal to :
(A) $-18$
(B) $18$
(C) $\frac{-34}{9}$
(D) $\frac{34}{9}$
Let $A = [a_{ij}]$ be a square matrix of order $3$ such that $a_{ij} = \hat{j} - 2\hat{i}$. Then which of the following is true ?
(A) $a_{12} > 0$
(B) all $a_{ij} < 0$
(C) $a_{13} + a_{31} = -6$
(D) $a_{23} > a_{32}$
The unit vector perpendicular to the vectors $\hat{i} - \hat{j}$ and $\hat{i} + \hat{j}$ is
(A) $\hat{k}$
(B) $-\hat{k} + \hat{j}$
(C) $\frac{\hat{i} - \hat{j}}{\sqrt{2}}$
(D) $\frac{\hat{i} + \hat{j}}{\sqrt{2}}$
The projection vector of vector $\vec{a}$ on vector $\vec{b}$ is
(A) $\left$ \frac{\vec{a} \cdot \vec{b}}{|\vec{b}|^2} \right$ \vec{b}$
(B) $\frac{\vec{a} \cdot \vec{b}}{|\vec{b}|}$
(C) $\frac{\vec{a} \cdot \vec{b}}{|\vec{a}|}$
(D) $\left$ \frac{\vec{a} \cdot \vec{b}}{|\vec{a}|^2} \right$ \vec{b}$
Let $\vec{p}$ and $\vec{q}$ be two unit vectors and $\alpha$ be the angle between them. Then $$\vec{p} + \vec{q}$$ will be a unit vector for what value of $\alpha$ ?
(A) $\frac{\pi}{4}$
(B) $\frac{\pi}{3}$
(C) $\frac{\pi}{2}$
(D) $\frac{2\pi}{3}$
(a) A vector $\vec{a}$ makes equal angles with all the three axes. If the magnitude of the vector is $5\sqrt{3}$ units, then find $\vec{a}$.
OR
(b) If $\vec{\alpha}$ and $\vec{\beta}$ are position vectors of two points P and Q respectively, then find the position vector of a point R in QP produced such that $\text{QR} = \frac{3}{2} \text{QP}$.
If $\vec{a}$ and $\vec{b}$ are two non-collinear vectors, then find $x$, such that $\vec{a} = (x - 2) \vec{a} + \vec{b}$ and $\vec{b} = (3 + 2x) \vec{a} - 2\vec{b}$ are collinear.
The values of $x$ for which the angle between the vectors $\vec{a} = 2x^2 \hat{i} + 4x \hat{j} + \hat{k}$ and $\vec{b} = 7\hat{i} - 2\hat{j} + x\hat{k}$ is obtuse, is :
(A) $0$ or $\frac{1}{2}$
(B) $x > \frac{1}{2}$
(C) $\left( 0, \frac{1}{2} \right)$
(D) $\left[ 0, \frac{1}{2} \right]$
If $\vec{PQ} \times \vec{PR} = 4\hat{i} + 8\hat{j} - 8\hat{k}$, then the area $(\Delta PQR)$ is
(A) 2 sq units
(B) 4 sq units
(C) 6 sq units
(D) 12 sq units
Let $\vec{p} = 2\hat{i} - 3\hat{j} - \hat{k}, \vec{q} = -3\hat{i} + 4\hat{j} + \hat{k}$ and $\vec{r} = \hat{i} + \hat{j} + 2\hat{k}$. Express $\vec{r}$ in the form of $\vec{r} = \lambda\vec{p} + \mu\vec{q}$ and hence find the values of $\lambda$ and $\mu$.
(a) A vector $\vec{a}$ makes equal angles with all the three axes. If the magnitude of the vector is $5\sqrt{3}$ units, then find $\vec{a}$.
OR
(b) If $\vec{\alpha}$ and $\vec{\beta}$ are position vectors of two points $P$ and $Q$ respectively, then find the position vector of a point $R$ in $QP$ pr...
The number of vector(s) of unit length perpendicular to the vectors $\vec{a} = 2\hat{i} + \hat{j} + 2\hat{k}$ and $\vec{b} = \hat{j} + \hat{k}$ is (are) :
(A) one
(B) two
(C) three
(D) infinite
Assertion (A) : The vectors $\vec{a} = 4\hat{i} + \hat{j} - \hat{k}$ and $\vec{b} = -2\hat{i} + 3\hat{j} - 5\hat{k}$ are mutually perpendicular vectors.\nReason (R) : Two vectors $\vec{a}$ and $\vec{b}$ are perpendicular to each other, if $\vec{a} \cdot \vec{b} = 0$.
If $|\vec{a} + \vec{b}| = 60$, $|\vec{a} - \vec{b}| = 40$ and $|\vec{b}| = 46$, then find $|\vec{a}|$.
If $\vec{a} = \hat{i} + 2\hat{j} + \hat{k}$, $\vec{b} = 2\hat{i} + \hat{j}$ and $\vec{c} = 3\hat{i} - 4\hat{j} - 5\hat{k}$, then find a unit vector perpendicular to both the vectors $$\vec{a} - \vec{b}$$ and $$\vec{c} - \vec{b}$$.
Find the shortest distance between the lines given by
$\vec{r} = $4\hat{i} - \hat{j} + 2\hat{k}$ + \lambda$\hat{i} + 2\hat{j} - 3\hat{k}$$ and
$\vec{r} = $2\hat{i} + \hat{j} - \hat{k}$ + \mu$3\hat{i} + 2\hat{j} - 4\hat{k}$$
Find the coordinates of the foot of the perpendicular and the length of the perpendicular drawn from the point P(5, 4, 2) to the line $\vec{r} = -\hat{i} + 3\hat{j} + \hat{k} + \lambda$2\hat{i} + 3\hat{j} - \hat{k}$$.
If the sides AB and AC of Δ ABC are represented by vectors $\hat{j} + \hat{k}$ and $3\hat{i} - \hat{j} + 4\hat{k}$ respectively, then the length of the median through A on BC is : (A) $2\sqrt{2}$ units (B) $\sqrt{18}$ units (C) $\frac{\sqrt{34}}{2}$ units (D) $\frac{\sqrt{48}}{2}$ units
(b) Find the equation of a line in vector and cartesian form which passes through the point $(1, 2, -4)$ and is perpendicular to the lines $\frac{x - 8}{3} = \frac{y + 19}{-16} = \frac{z - 10}{7}$, and $\vec{r} = 15\hat{i} + 29\hat{j} + 5\hat{k} + \mu$3\hat{i} + 8\hat{j} - 5\hat{k}$$.
The respective values of $|\vec{a}|$ and $|\vec{b}|$, if given $$\vec{a} - \vec{b}$ \cdot $\vec{a} + \vec{b}$ = 512$ and $|\vec{a}| = 3 |\vec{b}|$, are :
(A) $48$ and $16$
(B) $3$ and $1$
(C) $24$ and $8$
(D) $6$ and $2$
Let $\vec{a}$ be a position vector whose tip is the point $(2, -3)$. If $\vec{AB} = \vec{a}$, where coordinates of A are $(-4, 5)$, then the coordinates of B are :
(A) $(-2, -2)$
(B) $(2, -2)$
(C) $(-2, 2)$
(D) $(2, 2)$
The respective values of $|\vec{a}|$ and $|\vec{b}|$, if given $$\vec{a} - \vec{b}$ \cdot $\vec{a} + \vec{b}$ = 512$ and $|\vec{a}| = 3 |\vec{b}|$, are :
(A) $48 ext{ and } 16$
(B) $3 ext{ and } 1$
(C) $24 ext{ and } 8$
(D) $6 ext{ and } 2$
(a) Find a vector of magnitude $5$ which is perpendicular to both the vectors $3\hat{i} - 2\hat{j} + \hat{k}$ and $4\hat{i} + 3\hat{j} - 2\hat{k}$.\nOR\n(b) Let $\vec{a}$, $\vec{b}$ and $\vec{c}$ be three vectors such that $\vec{a} \cdot \vec{b} = \vec{a} \cdot \vec{c}$ and $\vec{a} \times \vec{b} =...
(a) If $\vec{a} + \vec{b} + \vec{c} = \vec{0}$ such that $|\vec{a}| = 3$, $|\vec{b}| = 5$, $|\vec{c}| = 7$, then find the angle between $\vec{a}$ and $\vec{b}$. \
OR \
(b) If $\vec{a}$ and $\vec{b}$ are unit vectors inclined with each other at an angle $\theta$, then prove that $\frac{1}{2} |\vec{a}...
The respective values of $|\vec{a}|$ and $|\vec{b}|$, if given
$$\vec{a} - \vec{b}$ \cdot $\vec{a} + \vec{b}$ = 512$ and $|\vec{a}| = 3 |\vec{b}|$, are :
(A) $48 \text{ and } 16$
(B) $3 \text{ and } 1$
(C) $24 \text{ and } 8$
(D) $6 \text{ and } 2$
Determine if the lines $\vec{r} = $\hat{i} + \hat{j} - \hat{k}$ + \lambda $3\hat{i} - \hat{j}$$ and $\vec{r} = $4\hat{i} - \hat{k}$ + \mu $2\hat{i} + 3\hat{k}$$ intersect with each other.
(a) Find the distance of the point $\text{P}(2, 4, -1)$ from the line $\frac{x + 5}{1} = \frac{y + 3}{4} = \frac{z - 6}{-9}$.
OR
(b) Let the position vectors of the points $\text{A}$, $\text{B}$ and $\text{C}$ be $3\hat{i} - \hat{j} - 2\hat{k}$, $\hat{i} + 2\hat{j} - \hat{k}$ and $\hat{i} + 5\hat{...
The vector with terminal point A $(2, -3, 5)$ and initial point B $(3, -4, 7)$ is :
(A) $\hat{i} - \hat{j} + 2\hat{k}$
(B) $\hat{i} + \hat{j} + 2\hat{k}$
(C) $-\hat{i} - \hat{j} - 2\hat{k}$
(D) $-\hat{i} + \hat{j} - 2\hat{k}$
In the given figure, $ABCD$ is a parallelogram. If $\vec{AB} = 2\hat{i} - 4\hat{j} + 5\hat{k}$ and $\vec{DB} = 3\hat{i} - 6\hat{j} + 2\hat{k}$, then find $\vec{AD}$ and hence find the area of parallelogram $ABCD$.
The value of $\alpha$ if the angle between $\vec{p} = 2\alpha^2 \hat{i} - 3\alpha \hat{j} + \hat{k}$ and $\vec{q} = \hat{i} + \hat{j} + \alpha \hat{k}$ is obtuse, is (A) $R - [0, 1]$ (B) $(0, 1)$ (C) $[0, \infty)$ (D) $[1, \infty)$
If $|\vec{a}| = 3, |\vec{b}| = 4$ and $|\vec{a} + \vec{b}| = 5$, then $|\vec{a} - \vec{b}| = $ (A) $3$ (B) $4$ (C) $5$ (D) $8$
(a) If vectors $\vec{a} = 2\hat{i} + 2\hat{j} + 3\hat{k}$, $\vec{b} = -\hat{i} + 2\hat{j} + \hat{k}$ and $\vec{c} = 3\hat{i} + \hat{j}$ are such that $\vec{b} + \lambda\vec{c}$ is perpendicular to $\vec{a}$, then find the value of $\lambda$.
(b) A person standing at $O(0, 0, 0)$ is watching an aeroplane which is at the coordinate point $A(4, 0, 3)$. At the same time he saw a bird at the coordinate point $B(0, 0, 1)$. Find the angles which $\vec{BA}$ makes with the $x, y$ and $z$ axes.
The two co-initial adjacent sides of a parallelogram are $2\hat{i} - 4\hat{j} - 5\hat{k}$ and $2\hat{i} + 2\hat{j} + 3\hat{k}$. Find its diagonals and use them to find the area of the parallelogram.
An ant is moving along the vector $\vec{l_1} = \hat{i} - 2\hat{j} + 3\hat{k}$. Few sugar crystals are kept along the vector $\vec{l_2} = 3\hat{i} - 2\hat{j} + \hat{k}$ which is inclined at an angle $\theta$ with the vector $\vec{l_1}$. Then find the angle $\theta$. Also find the scalar projection of...
Find the vector and the cartesian equation of the line that passes through $(-1, 2, 7)$ and is perpendicular to the lines $\vec{r} = 2\hat{i} + \hat{j} - 3\hat{k} + \lambda$\hat{i} + 2\hat{j} + 5\hat{k}$$ and $\vec{r} = 3\hat{i} + 3\hat{j} - 7\hat{k} + \mu$3\hat{i} - 2\hat{j} + 5\hat{k}$$.
Find the shortest distance between the lines $l_1$ and $l_2$ whose vector equations are $\vec{r} = $-\hat{i} - \hat{j} - \hat{k}$ + \lambda$7\hat{i} - 6\hat{j} + \hat{k}$$ and $\vec{r} = $3\hat{i} + 5\hat{j} + 7\hat{k}$ + \mu$\hat{i} - 2\hat{j} + \hat{k}$$ where $\lambda$ and $\mu$ are parameters.
For any two vectors $\vec{a}$ and $\vec{b}$, which of the following statements is always true?
(A) $\vec{a} \cdot \vec{b} \geq |\vec{a}| |\vec{b}|$
(B) $\vec{a} \cdot \vec{b} = |\vec{a}| |\vec{b}|$
(C) $\vec{a} \cdot \vec{b} \leq |\vec{a}| |\vec{b}|$
(D) $\vec{a} \cdot \vec{b} < |\vec{a}| |\vec{...
If vectors $\vec{a}$, $\vec{b}$ and $2\vec{a} + 3\vec{b}$ are unit vectors, then find the angle between $\vec{a}$ and $\vec{b}$.
Find the value of p for which the lines $\vec{r} = \lambda \hat{i} + $2\lambda + 1$\hat{j} + $3\lambda + 2$\hat{k}$ and $\vec{r} = \hat{i} - 3\mu\hat{j} + $p\mu + 7$\hat{k}$ are perpendicular to each other and also intersect. Also, find the point of intersection of the given lines.
Assertion $(A) :$ $$\vec{b} \cdot \vec{c}$ \vec{a}$ is a scalar quantity.
Reason $(R) :$ Dot product of two vectors is a scalar quantity.
Let the vectors $\vec{a}$ and $\vec{b}$ be such that $|\vec{a}| = \sqrt{3}$ and $|\vec{b}| = \frac{2}{\sqrt{3}}$, then $\vec{a} \times \vec{b}$ is a unit vector, if the angle between $\vec{a}$ and $\vec{b}$ is :
(A) $\frac{\pi}{3}$
(B) $\frac{\pi}{4}$
(C) $\frac{\pi}{6}$
(D) $\frac{\pi}{2}$
If $\vec{a} = 2\hat{i} - 2\hat{j} + \hat{k}$, $\vec{b} = \hat{i} + 2\hat{j} - 3\hat{k}$ and $\vec{c} = 2\hat{i} - \hat{j} + 4\hat{k}$, then the projection of $$\vec{c} - \vec{b}$$ along $\vec{a}$ is :
(A) $\frac{15}{2}$
(B) 5
(C) $\frac{2}{3}$
(D) 1
If $\vec{a}, \vec{b}$ and $\vec{c}$ are unit vectors such that $\vec{a} + \vec{b} + \vec{c} = \vec{0}$, then $$\vec{a} \cdot \vec{b} + \vec{b} \cdot \vec{c} + \vec{c} \cdot \vec{a}$$ is equal to :
(A) $\frac{3}{2}$
(B) $\frac{1}{2}$
(C) $-\frac{1}{2}$
(D) $-\frac{3}{2}$
A cricket match is organised between two clubs P and Q for which a team from each club is chosen. Remaining players of club P and club Q are respectively sitting along the lines AB and CD, where the points are A(3, 4, 0), B(5, 3, 3), C(6, -4, 1) and D(13, -5, -4). Based on the above, answer the foll...
(ii) Write a unit vector in the direction of $\vec{CD}$.
(iii) (a) Find the angle between vectors $\vec{AB}$ and $\vec{CD}$.
(iii) (b) Write a vector perpendicular to both $\vec{AB}$ and $\vec{CD}$.
Let $\vec{a}$ and $\vec{b}$ be two non-zero vectors. Prove that $|\vec{a} \times \vec{b}| \leq |\vec{a}| |\vec{b}|$. State the condition under which equality holds, i.e., $|\vec{a} \times \vec{b}| = |\vec{a}| |\vec{b}|$.
If $\vec{a}$, $\vec{b}$ and $\vec{c}$ are three unit vectors such that $\vec{a} + \vec{b} - \vec{c} = \vec{0}$, find the angle between vectors $\vec{a}$ and $\vec{c}$.
Assertion (A) : For any non-zero unit vector $\vec{a}$, $\vec{a} .$-\vec{a}$ =$-\vec{a}$ .\vec{a} = -1$.
Reason (R) : Angle between $\vec{a}$ and $$-\vec{a}$$ is $\frac{\pi}{2}$.
$\vec{a}$, $\vec{b}$ and $\vec{c}$ are three mutually perpendicular unit vectors. If $\theta$ is the angle between $\vec{a}$ and $$2\vec{a} + 3\vec{b} + 6\vec{c}$$, find the value of $\cos \theta$.
The vector equation of a line passing through the point $(1, -1, 0)$ and parallel to Y-axis is :
(A) $\vec{r} = \hat{i} - \hat{j} + \lambda$\hat{i} - \hat{j}$$
(B) $\vec{r} = \hat{i} - \hat{j} + \lambda\hat{j}$
(C) $\vec{r} = \hat{i} - \hat{j} + \lambda\hat{k}$
(D) $\vec{r} = \lambda\hat{j}$
(ii) Find a unit vector in the direction of $\vec{\text{DA}}$ .
OR (iii) What is the projection of vector $\vec{\text{DV}}$ on vector $\vec{\text{DA}}$ ?
A vector perpendicular to the line $\vec{r} = \hat{i} + \hat{j} - \hat{k} + \lambda$3\hat{i} - \hat{j}$$ is :
(A) $5\hat{i} + \hat{j} + 6\hat{k}$
(B) $\hat{i} + 3\hat{j} + 5\hat{k}$
(C) $2\hat{i} - 2\hat{j}$
(D) $9\hat{i} - 3\hat{j}$
The Cartesian equation of a line passing through the point with position vector $\vec{a} = \hat{i} - \hat{j}$ and parallel to the line $\vec{r} = \hat{i} + \hat{k} + \mu$2\hat{i} - \hat{j}$$, is
(A) $\frac{x - 2}{1} = \frac{y + 1}{0} = \frac{z}{1}$
(B) $\frac{x - 1}{2} = \frac{y + 1}{-1} = \frac{z}{...
The Cartesian equation of the line passing through the point $(1, -3, 2)$ and parallel to the line $\vec{r}=2\hat{i}-\hat{k}+\lambda$\hat{i}+\hat{j}+2\hat{k}$$ is:
If the vectors $\vec{a}$, $\vec{b}$ and $\vec{c}$ represent the three sides of a triangle, then show that $\vec{a} \times \vec{b} = \vec{b} \times \vec{c} = \vec{c} \times \vec{a}$.
The vectors $\vec{a} = 2\hat{i} - 4\hat{j} + \lambda\hat{k}$ and $\vec{b} = 3\hat{i} - 6\hat{j} + \hat{k}$ are collinear if value of $\lambda$ is :
(A) $-30$
(B) 30
(C) $\frac{3}{2}$
(D) $\frac{2}{3}$
The vector equation of the line passing through the origin and perpendicular to the lines $\frac{x}{1} = \frac{y}{0} = \frac{z}{0}$ and $\frac{x}{0} = \frac{y}{0} = \frac{z}{1}$ is :
(A) $\vec{r} = \lambda\hat{i}$
(B) $\vec{r} = \lambda\hat{j}$
(C) $\vec{r} = \lambda\hat{k}$
(D) $\vec{r} = \lambda$\...
Find all vectors of magnitude $8\sqrt{14}$ units that are perpendicular to the vectors $2\hat{i} - \hat{k}$ and $2\hat{j} + 3\hat{k}$.
The value of $$\hat{i} \times \hat{j}$ \cdot \hat{j} + $\hat{j} \times \hat{i}$ \cdot \hat{k}$ is :
If $\vec{a} + \vec{b} = \hat{i}$ and $\vec{a} = 2\hat{i} - 2\hat{j} + 2\hat{k}$, then $| \vec{b} |$ equals :
(a) Position vectors of the points A, B and C as shown in the figure below are $\vec{a}$, $\vec{b}$ and $\vec{c}$ respectively. [Figure showing points A, B, C on a line]. If $\vec{AC} = \frac{5}{4} \vec{AB}$, express $\vec{c}$ in terms of $\vec{a}$ and $\vec{b}$.
If the vector $\hat{i} - b\hat{j} + \hat{k}$ is equally inclined to the coordinate axes, then the value of b is :
(a) -1
(b) 1
(c) $-\sqrt{3}$
(d) $-\frac{1}{\sqrt{3}}$
For two non-zero vectors $\vec{a}$ and $\vec{b}$, if $|\vec{a} - \vec{b}| = |\vec{a} + \vec{b}|$, then find the angle between $\vec{a}$ and $\vec{b}$.
For what value of $\lambda$, the projection of vector $\hat{i} + \lambda\hat{j}$ on vector $\hat{i} - \hat{j}$ is $\sqrt{2}$ ?
(a) -1
(b) 1
(c) 0
(d) 3
The lines $\vec{r} = \hat{i} + \hat{j} - \hat{k} + \lambda $2\hat{i} + 3\hat{j} - 6\hat{k}$$ and $\vec{r} = 2\hat{i} - \hat{j} - \hat{k} + \mu $6\hat{i} + 9\hat{j} - 18\hat{k}$$; (where $\lambda \& \mu$ are scalars) are (a) coincident (b) skew (c) intersecting (d) parallel
$ABCD$ is a rhombus whose diagonals intersect at $E$. Then $\vec{EA} + \vec{EB} + \vec{EC} + \vec{ED}$ equals to (a) $\vec{0}$ (b) $\vec{AD}$ (c) $2\vec{BD}$ (d) $2\vec{AD}$
If $\vec{a} = 4\hat{i} + 6\hat{j}$ and $\vec{b} = 3\hat{j} + 4\hat{k}$, then the vector form of the component of $\vec{a}$ along $\vec{b}$ is
(a) $\frac{18}{5}$3\hat{j} + 4\hat{k}$$
(b) $\frac{18}{25}$3\hat{j} + 4\hat{k}$$
(c) $\frac{18}{5}$3\hat{i} + 4\hat{k}$$
(d) $\frac{18}{25}$4\hat{i} + 6\hat{j...
The value of $\lambda$ for which two vectors $2\hat{i} - \hat{j} + 2\hat{k}$ and $3\hat{i} + \lambda\hat{j} + \hat{k}$ are perpendicular is
(a) 2
(b) 4
(c) 6
(d) 8
An aeroplane is flying along the line $\vec{r} = \lambda$\hat{i} - \hat{j} + \hat{k}$$; where $\lambda$ is a scalar and another aeroplane is flying along the line $\vec{r} = \hat{i} - \hat{j} + \mu$-2\hat{j} + \hat{k}$$; where $\mu$ is a scalar. At what points on the lines should they reach, so that...
Find the coordinates of the image of the point $(1, 6, 3)$ with respect to the line $\vec{r} = $\hat{j} + 2\hat{k}$ + \lambda$\hat{i} + 2\hat{j} + 3\hat{k}$$; where $\lambda$ is a scalar. Also, find the distance of the image from the $y-\text{axis}$.
Read the following passage and answer the questions given below: Teams A, B, C went for playing a tug of war game. Teams A, B, C have attached a rope to a metal ring and is trying to pull the ring into their own area. Team A pulls with force $\vec{F}_1 = 6\hat{i} + 0\hat{j} \text{ kN}$, Team B pulls...
Position vector of the mid-point of line segment AB is $3\hat{i} + 2\hat{j} - 3\hat{k}$. If position vector of the point A is $2\hat{i} + 3\hat{j} - 4\hat{k}$, then position vector of the point B is
(A) $\frac{5\hat{i}}{2} + \frac{5\hat{j}}{2} - \frac{7\hat{k}}{2}$
(B) $4\hat{i} + \hat{j} - 2\hat{k}...
Projection of vector $2\hat{i} + 3\hat{j}$ on the vector $3\hat{i} - 2\hat{j}$ is
(A) $0$
(B) $12$
(C) $\frac{12}{\sqrt{13}}$
(D) $\frac{-12}{\sqrt{13}}$
If in $\Delta ABC$, $\vec{BA} = 2\vec{a}$ and $\vec{BC} = 3\vec{b}$, then $\vec{AC}$ is
(A) $2\vec{a} + 3\vec{b}$
(B) $2\vec{a} - 3\vec{b}$
(C) $3\vec{b} - 2\vec{a}$
(D) $-2\vec{a} - 3\vec{b}$
If $|\vec{a} \times \vec{b}| = \sqrt{3}$ and $\vec{a} \cdot \vec{b} = -3$, then angle between $\vec{a}$ and $\vec{b}$ is
(A) $\frac{2\pi}{3}$
(B) $\frac{\pi}{6}$
(C) $\frac{\pi}{3}$
(D) $\frac{5\pi}{6}$
If the angle between $\vec{a}$ and $\vec{b}$ is $\frac{\pi}{3}$ and $|\vec{a} \times \vec{b}| = 3\sqrt{3}$, then the value of $\vec{a} \cdot \vec{b}$ is
(A) 9
(B) 3
(C) $\frac{1}{9}$
(D) $\frac{1}{3}$
The position vectors of three consecutive vertices of a parallelogram ABCD are $A$4\hat{i} + 2\hat{j} - 6\hat{k}$$, $B$5\hat{i} - 3\hat{j} + \hat{k}$$ and $C$12\hat{i} + 4\hat{j} + 5\hat{k}$$. The position vector of D is given by
(A) $-3\hat{i} - 5\hat{j} - 10\hat{k}$
(B) $21\hat{i} + 3\hat{j}$
(C) ...
If points A, B and C have position vectors $2\hat{i}$, $\hat{j}$ and $2\hat{k}$ respectively, then show that $\Delta \text{ABC}$ is an isosceles triangle.
Let $\vec{a}$ and $\vec{b}$ be two unit vectors and $\theta$ is the angle between them. $\vec{a} + \vec{b}$ is a unit vector, if:
(a) $\theta = \frac{\pi}{3}$
(b) $\theta = \frac{\pi}{4}$
(c) $\theta = \frac{\pi}{2}$
(d) $\theta = \frac{2\pi}{3}$
If $$2\hat{i} + 6\hat{j} - 22\hat{k}$ \times $\hat{i} + \lambda\hat{j} + \mu\hat{k}$ = \vec{0}$, then $\lambda - \mu$ is equal to:
(a) $-8$
(b) $-14$
(c) 14
(d) 8
If the position vectors of two points A and B are $\hat{i} + 2\hat{j} - 3\hat{k}$ and $-\hat{i} - 2\hat{j} + \hat{k}$ respectively, then the direction cosines of the vector $\vec{BA}$ are:
(a) $\frac{2}{6}, -\frac{4}{6}, -\frac{4}{6}$
(b) $\frac{1}{3}, \frac{2}{3}, -\frac{2}{3}$
(c) $\frac{1}{\sqrt{...
Assertion (A) : The vector equation of a line passing through the points A(-1, 0, 2) and B(3, 4, 6) is $\vec{r} = -\hat{i} + 2\hat{k} + \lambda$\hat{i} + \hat{j} + \hat{k}$$. Reason (R) : The equation of a line passing through a point with position vector $\vec{a}$ and parallel to a vector $\vec{b}$...
Find the projection of the vector $7\hat{i} - \hat{j} + 8\hat{k}$ on the vector $\hat{i} + 2\hat{j} + 2\hat{k}$.
(a) In a parallelogram ABCD, the sides AB and AD are represented by the vectors $2\hat{i} - 4\hat{j} + 5\hat{k}$ and $\hat{i} - 2\hat{j} - 3\hat{k}$ respectively. Find the unit vector parallel to its diagonal $\vec{AC}$.
(b) Find the angle between the pair of lines given by $\vec{r} = \hat{i} + 2\hat{j} - 2\hat{k} + \lambda$\hat{i} - 2\hat{j} - 2\hat{k}$$ and $\vec{r} = 3\hat{i} - 5\hat{j} + \hat{k} + \mu$3\hat{i} + 2\hat{j} - 6\hat{k}$$.
(a) Find the shortest distance between the lines whose vector equations are : $\vec{r} = \hat{i} + 2\hat{j} - 4\hat{k} + \lambda$2\hat{i} + 3\hat{j} + 6\hat{k}$$ and $\vec{r} = 3\hat{i} - 3\hat{j} - 5\hat{k} + \mu$- 2\hat{i} + 3\hat{j} + 8\hat{k}$$ OR (b) Find the foot of the perpendicular drawn fro...
The value of $\lambda$ for which the angle between the lines
$\vec{r} = \hat{i} + \hat{j} + \hat{k} + p$2\hat{i} + \hat{j} + 2\hat{k}$$ and
$\vec{r} = (1 + q)\hat{i} + $1 + q\lambda$\hat{j} + (1 + q)\hat{k}$ is $\frac{\pi}{2}$ is :
(a) $-4$
(b) $4$
(c) $2$
(d) $-2$
If $\vec{r} = 3\hat{i} - 2\hat{j} + 6\hat{k}$, find the value of $$\vec{r} \times \hat{j}$ \cdot $\vec{r} \times \hat{k}$ - 12$.
The direction cosines of vector $\vec{\text{BA}}$, where coordinates of A and B are $(1, 2, -1)$ and $(3, 4, 0)$ respectively, are :
(a) $-2, -2, -1$
(b) $-\frac{2}{3}, -\frac{2}{3}, -\frac{1}{3}$
(c) $2, 2, 1$
(d) $\frac{2}{3}, \frac{2}{3}, \frac{1}{3}$
$\vec{\text{a}}$ and $\vec{\text{b}}$ are two non-zero vectors such that the projection of $\vec{\text{a}}$ on $\vec{\text{b}}$ is 0. The angle between $\vec{\text{a}}$ and $\vec{\text{b}}$ is :
(a) $\frac{\pi}{2}$
(b) $\pi$
(c) $\frac{\pi}{4}$
(d) 0
In $\Delta \text{ABC}$, $\vec{\text{AB}} = \hat{\text{i}} + \hat{\text{j}} + 2\hat{\text{k}}$ and $\vec{\text{AC}} = 3\hat{\text{i}} - \hat{\text{j}} + 4\hat{\text{k}}$. If D is mid-point of BC, then vector $\vec{\text{AD}}$ is equal to :
(a) $4\hat{\text{i}} + 6\hat{\text{k}}$
(b) $2\hat{\text{i}}...
If $\vec{r} = 3\hat{i} - 2\hat{j} + 6\hat{k}$, find the value of $$\vec{r} \times \hat{j}$ . $\vec{r} \times \hat{k}$ - 12$.
If the angle between the vectors $\vec{a}$ and $\vec{b}$ is $\frac{\pi}{4}$ and $| \vec{a} \times \vec{b} | = 1$, then $\vec{a} \cdot \vec{b}$ is equal to
(a) $-1$
(b) $1$
(c) $\frac{1}{\sqrt{2}}$
(d) $\sqrt{2}$
If $\vec{a} = 4\hat{i} - \hat{j} + \hat{k}$ and $\vec{b} = 2\hat{i} - 2\hat{j} + \hat{k}$, then find a unit vector along the vector $\vec{a} \times \vec{b}$.
The sine of the angle between the vectors $\vec{a} = 3\hat{i} + \hat{j} + 2\hat{k}$ and $\vec{b} = \hat{i} + \hat{j} + 2\hat{k}$ is:
(a) $\sqrt{\frac{5}{21}}$
(b) $\frac{5}{\sqrt{21}}$
(c) $\sqrt{\frac{3}{21}}$
(d) $\frac{4}{\sqrt{21}}$
Questions number 19 and 20 are Assertion and Reason based questions carrying 1 mark each. Two statements are given, one labelled Assertion (A) and the other labelled Reason (R). Select the correct answer from the codes (a), (b), (c) and (d) as given below.
(a) Both Assertion (A) and Reason (R) are t...
The magnitude of the vector $6 \hat{i} - 2 \hat{j} + 3 \hat{k}$ is
(a) 1
(b) 5
(c) 7
(d) 12
A unit vector $\hat{a}$ makes equal but acute angles on the co-ordinate axes. The projection of the vector $\hat{a}$ on the vector $\vec{b} = 5\hat{i} + 7\hat{j} - \hat{k}$ is
(a) $\frac{11}{15}$
(b) $\frac{11}{5\sqrt{3}}$
(c) $\frac{4}{5}$
(d) $\frac{3}{5\sqrt{3}}$
Find the angle between the following two lines :
$\vec{r} = 2\hat{i} - 5\hat{j} + \hat{k} + \lambda$3\hat{i} + 2\hat{j} + 6\hat{k}$$ ;
$\vec{r} = 7\hat{i} - 6\hat{k} + \mu$\hat{i} + 2\hat{j} + 2\hat{k}$$
The vector equation of a line which passes through the point $(2, -4, 5)$ and is parallel to the line $\frac{x+3}{3} = \frac{4-y}{2} = \frac{z+8}{6}$ is :
(a) $\vec{r} = $-2\hat{i} + 4\hat{j} - 5\hat{k}$ + \lambda$3\hat{i} + 2\hat{j} + 6\hat{k}$$
(b) $\vec{r} = $2\hat{i} - 4\hat{j} + 5\hat{k}$ + \la...
If three non-zero vectors are $\vec{a}$, $\vec{b}$ and $\vec{c}$ such that $\vec{a} \cdot \vec{b} = \vec{a} \cdot \vec{c}$ and $\vec{a} \times \vec{b} = \vec{a} \times \vec{c}$, then show that $\vec{b} = \vec{c}$.
If ABCD is a parallelogram and $\vec{AC}$ and $\vec{BD}$ are its diagonals, then $\vec{AC} + \vec{BD}$ is :
(a) $2\vec{DA}$ (b) $2\vec{AB}$ (c) $2\vec{BC}$ (d) $2\vec{BD}$
The vector equation of a line which passes through the point (1, - 2, 3) and is parallel to the vector $3\hat{i} - 2\hat{j} + 4\hat{k}$ is : (a) $\vec{r} = $-\hat{i} + 2\hat{j} + 3\hat{k}$ + \lambda$3\hat{i} - 2\hat{j} + 4\hat{k}$$ (b) $\vec{r} = $-3\hat{i} + 2\hat{j} - 4\hat{k}$ + \lambda$\hat{i} -...
Let P and Q be two points with position vectors $\vec{a} - 2\vec{b}$ and $2\vec{a} + \vec{b}$ respectively. The position vector of a point which divides the join of P and Q externally in the ratio $3:2$ is :
(a) $4\vec{a} + 7\vec{b}$
(b) $\frac{8\vec{a} + 7\vec{b}}{5}$
(c) $4\vec{a} - 7\vec{b}$
(d) ...
For the vectors $\vec{a} = \hat{i} - 2\hat{j} + 3\hat{k}$ and $\vec{b} = 2\hat{i} + 3\hat{j} - 5\hat{k}$, verify that the angle between $\vec{a}$ and $\vec{a} \times \vec{b}$ is $\frac{\pi}{2}$.
Find a vector of magnitude 6, which is perpendicular to each of the vectors $\vec{a} + \vec{b}$ and $\vec{a} - \vec{b}$, where $\vec{a} = \hat{i} + \hat{j} + \hat{k}$ and $\vec{b} = \hat{i} + 2\hat{j} + 3\hat{k}$.
If a unit vector $\vec{a}$ makes angles $\frac{\pi}{3}$ with $\hat{i}$, $\frac{\pi}{4}$ with $\hat{j}$ and an acute angle $\theta$ with $\hat{k}$, then the value of $\theta$ is: (a) $\frac{\pi}{3}$ (b) $\frac{\pi}{4}$ (c) $\frac{\pi}{2}$ (d) $\frac{\pi}{6}$
The position vectors of two points $A$ and $B$ are $$2\vec{a} + \vec{b}$$ and $$\vec{a} - 3\vec{b}$$ respectively. The position vector of a point $C$ which divides $AB$ externally in the ratio $1 : 2$ is :
(a) $-3\vec{a} - 5\vec{b}$
(b) $-7\vec{b}$
(c) $\frac{1}{3}$5\vec{a} - \vec{b}$$
(d) $$3\v...
The value of $\lambda$ for which the points $A$, $B$ and $C$ having position vectors $$3\hat{i} - 2\hat{j} + 4\hat{k}$$, $$\hat{i} + \lambda\hat{j} + \hat{k}$$ and $$-\hat{i} + 4\hat{j} - 2\hat{k}$$ respectively are collinear, is :
(a) $4$
(b) $1$
(c) $3$
(d) $2$
Assertion (A) : The vector equation of a line passing through the points $(1, 2, 3)$ and $(3, 0, 2)$ is $\vec{r} = \hat{i} + 2\hat{j} + 3\hat{k} + \lambda$2\hat{i} - 2\hat{j} - \hat{k}$$.
Reason (R) : Equation of a line passing through two points with position vectors $\vec{a}$ and $\vec{b}$ is $\ve...
Express the vector $\vec{a} = 5\hat{i} - 2\hat{j} + 5\hat{k}$ as the sum of two vectors such that one is parallel to the vector $\vec{b} = $3\hat{i} + \hat{k}$$ and the other is perpendicular to $\vec{b}$.
(a) Show that the lines $\vec{r} = \hat{i} + 2\hat{j} + \hat{k} + \lambda$\hat{i} - \hat{j} + \hat{k}$$ and $\vec{r} = \hat{i} + \hat{j} + \hat{k} + \mu$\hat{i} - \hat{j} + 2\hat{k}$$ do not intersect. OR (b) Find the coordinates of the foot of perpendicular drawn from the point (3, -1, 11) to the l...
Find the vector equation of the plane passing through the intersection of the planes $\vec{r} \cdot $2\hat{i} + 2\hat{j} - 3\hat{k}$ = 7$ and $\vec{r} \cdot $2\hat{i} + 5\hat{j} + 3\hat{k}$ = 9$ and through the point $(2, 1, 3)$.
If $\vec{a} = 2\hat{i} + y\hat{j} + \hat{k}$ and $\vec{b} = \hat{i} + 2\hat{j} + 3\hat{k}$ are two vectors for which the vector $$\vec{a} + \vec{b}$$ is perpendicular to the vector $$\vec{a} - \vec{b}$$, then find all the possible values of y.
ABCD is a parallelogram such that $\vec{AC} = \hat{i} + \hat{j}$ and $\vec{BD} = 2\hat{i} + \hat{j} + \hat{k}$. Find $\vec{AB}$ and $\vec{AD}$. Also, find the area of the parallelogram ABCD.
The scalar projection of the vector $3\hat{i} - \hat{j} - 2\hat{k}$ on the vector $\hat{i} + 2\hat{j} - 3\hat{k}$ is
(a) $\frac{7}{\sqrt{14}}$ (b) $\frac{7}{14}$ (c) $\frac{6}{13}$ (d) $\frac{7}{2}$
If two vectors $\vec{a}$ and $\vec{b}$ are such that $|\vec{a}| = 2, |\vec{b}| = 3$ and $\vec{a} \cdot \vec{b} = 4$, then $|\vec{a} - 2\vec{b}|$ is equal to (a) $\sqrt{2}$ (b) $2\sqrt{6}$ (c) 24 (d) $2\sqrt{2}$
Assertion (A): The acute angle between the line $\vec{r} = \hat{i} + \hat{j} + 2\hat{k} + \lambda$\hat{i} - \hat{j}$$ and the x-axis is $\frac{\pi}{4}$.
Reason(R): The acute angle $\theta$ between the lines $\vec{r} = x_1\hat{i} + y_1\hat{j} + z_1\hat{k} + \lambda$a_1\hat{i} + b_1\hat{j} + c_1\hat{k...
If $\vec{a} = \hat{i} - \hat{j} + 7\hat{k}$ and $\vec{b} = 5\hat{i} - \hat{j} + \lambda\hat{k}$, then find the value of $\lambda$ so that the vectors $\vec{a} + \vec{b}$ and $\vec{a} - \vec{b}$ are orthogonal.
Find $|\vec{x}|$ if $$\vec{x} - \vec{a}$ \cdot $\vec{x} + \vec{a}$ = 12$, where $\vec{a}$ is a unit vector.
An insect is crawling along the line $\vec{r} = 6\hat{i} + 2\hat{j} + 2\hat{k} + \lambda$\hat{i} - 2\hat{j} + 2\hat{k}$$ and another insect is crawling along the line $\vec{r} = -4\hat{i} - \hat{k} + \mu$3\hat{i} - 2\hat{j} - 2\hat{k}$$. At what points on the lines should they reach so that the dist...
If $\vec{a}$ and $\vec{b}$ are two vectors such that $\vec{a} = \hat{i} - \hat{j} + \hat{k}$ and $\vec{b} = 2\hat{i} - \hat{j} - 3\hat{k}$, then find the vector $\vec{c}$, given that $\vec{a} \times \vec{c} = \vec{b}$ and $\vec{a} \cdot \vec{c} = 4$.
If $\vec{a}$, $\vec{b}$, $\vec{c}$ are three vectors such that $\vec{a} \cdot \vec{b} = \vec{a} \cdot \vec{c}$ and $\vec{a} \times \vec{b} = \vec{a} \times \vec{c}$, $\vec{a} \neq 0$, then show that $\vec{b} = \vec{c}$. OR If $|\vec{a}| = 3$, $|\vec{b}| = 5$, $|\vec{c}| = 4$ and $\vec{a} + \vec{b} +...
Find the shortest distance between the following lines : $\vec{r} = 3\hat{i} + 5\hat{j} + 7\hat{k} + \lambda$\hat{i} - 2\hat{j} + \hat{k}$$ and $\vec{r} = $-\hat{i} - \hat{j} - \hat{k}$ + \mu$7\hat{i} - 6\hat{j} + \hat{k}$$.
OR
If $\vec{a} = 2\hat{i} + 2\hat{j} + 3\hat{k}$, $\vec{b} = -\hat{i} + 2\hat{j} + \hat{k}$ and $\vec{c} = 3\hat{i} + \hat{j}$ are such that the vector $$\vec{a} + \lambda \vec{b}$$ is perpendicular to vector $\vec{c}$, then find the value of $\lambda$.
If $\vec{a}$, $\vec{b}$ and $\vec{c}$ are mutually perpendicular vectors of equal magnitude, then prove that the vector $$2\vec{a} + \vec{b} + 2\vec{c}$$ is equally inclined to both $\vec{a}$ and $\vec{c}$. Also, find the angle between $\vec{a}$ and $$2\vec{a} + \vec{b} + 2\vec{c}$$.
If $\vec{a}$ and $\vec{b}$ are two vectors of equal magnitude and $\alpha$ is the angle between them, then prove that $\frac{|\vec{a} + \vec{b}|}{|\vec{a} - \vec{b}|} = \cot \left$ \frac{\alpha}{2} \right$$.
(a) If $|\vec{a} \times \vec{b}|^2 + |\vec{a}.\vec{b}|^2 = 400$ and $|\vec{b}| = 5$, then find the value of $|\vec{a}|$. OR (b) Find all the possible vectors of magnitude $5\sqrt{3}$ which are equally inclined to the coordinate axes.
If $\vec{a} = 2\hat{i} - \hat{j} + \hat{k}$, $\vec{b} = \hat{i} + \hat{j} - 2\hat{k}$ and $\vec{c} = \hat{i} + 3\hat{j} - \hat{k}$ and the projection of vector $\vec{c} + \lambda \vec{b}$ on vector $\vec{a}$ is $2\sqrt{6}$, then find the value of $\lambda$.
If $\vec{a} = \hat{i} + \hat{j} + \hat{k}$, $\vec{a}.\vec{b} = 1$ and $\vec{a} \times \vec{b} = \hat{j} - \hat{k}$, then find $|\vec{b}|$
Two motorcycles A and B are running at the speed more than the allowed speed on the roads represented by the lines $\vec{r} = \lambda$\hat{i} + 2\hat{j} - \hat{k}$$ and $\vec{r} = $3\hat{i} + 3\hat{j}$ + \mu$2\hat{i} + \hat{j} + \hat{k}$$ respectively. Based on the above information, answer the foll...
(a) Find the distance between the following parallel lines :
$\vec{r} = $2\hat{i} + \hat{j} - \hat{k}$ + \lambda$\hat{i} + \hat{j} - \hat{k}$$
$\vec{r} = $\hat{i} - 2\hat{j} + \hat{k}$ + \mu$\hat{i} + \hat{j} - \hat{k}$$
OR
(b) Find the coordinates of the point where the line through the points $(-1...
Find the coordinates of the foot of the perpendicular drawn from the point $(1, 3, 4)$ to the plane $\vec{r} \cdot $2\hat{i} - \hat{j} + \hat{k}$ + 3 = 0$.
Find the distance between the following parallel lines : $\vec{r} = $2\hat{i} + \hat{j} - \hat{k}$ + \lambda$\hat{i} + \hat{j} - \hat{k}$$ and $\vec{r} = $\hat{i} - 2\hat{j} + \hat{k}$ + \mu$\hat{i} + \hat{j} - \hat{k}$$
Find the value of $\lambda$ for which the vectors $\vec{a} + \vec{b}$ and $\vec{a} - \vec{b}$ are perpendicular to each other, where $\vec{a} = 2\hat{i} - \hat{j} + \lambda\hat{k}$ and $\vec{b} = 3\hat{i} + \hat{j} - 2\hat{k}$.
Find the direction cosines of a line whose vector equation is given as $\vec{r} = $1 - \lambda$\hat{i} + $2\lambda - 1$\hat{j} + $2\lambda + 3$\hat{k}$.
For vectors $\vec{a} = \hat{i} + 2\hat{j} + 4\hat{k}$, $\vec{b} = - 2\hat{i} + 3\hat{j} + 2\hat{k}$ and $\vec{c} = 2\hat{i} + 2\hat{j} + 3\hat{k}$, determine (i) $\vec{b} \times \vec{c}$ and hence (ii) $\vec{a} \cdot $\vec{b} \times \vec{c}$$.
Find the shortest distance between the lines : $\vec{r} = $\hat{i} + 2\hat{j} + \hat{k}$ + \lambda$\hat{i} - \hat{j} + \hat{k}$$ and $\vec{r} = $2\hat{i} - \hat{j} - \hat{k}$ + \mu$2\hat{i} + \hat{j} + 2\hat{k}$$.