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 from A to C and B to C in such a way that $\vec{OA} = \vec{a}$, $\vec{OB} = \vec{b}$ and $\vec{OC} = 5\vec{a} - 2\vec{b}$ respectively. Based upon the above information, answer the following questions : (i) Complete the given figure to explain their entire movement plan along the respective vectors. (ii) Find vectors $\vec{AC}$ and $\vec{BC}$. (iii) (a) If $\vec{a} \cdot \vec{b} = 1$, distance of O to A is 1 km and that from O to B is 2 km, then find the angle between $\vec{OA}$ and $\vec{OB}$. Also, find $|\vec{a} \times \vec{b}|$. OR (iii) (b) If $\vec{a} = 2\hat{i} - \hat{j} + 4\hat{k}$ and $\vec{b} = \hat{j} - \hat{k}$, then find a unit vector perpendicular to $(\vec{a} + \vec{b})$ and $(\vec{a} - \vec{b})$.

Year	Paper Code	Set	Position
CBSE 2025	Series 65-7	Set 1	Question 36
CBSE 2025	Series 65-7	Set 2	Question 37
CBSE 2025	Series 65-7	Set 3	Question 38

• Step 1: Stating the fundamental formula / law statement (+1 Mark)
• Step 2: Intermediate mathematical substitution or proof steps (+1 Mark)
• Step 3: Correct final value or conclusion with mandatory SI units (+1 Mark)