Vector Algebra: 4 marks Questions (CBSE Class 12)

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 $\overrightarrow{OA}=\vec{a}$, $\overrightarrow{OB}=\vec{b}$ and $\overrightarrow{OC}=5\vec{a}-2\vec{b}$ 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. (1)
(ii) Find vectors $\overrightarrow{AC}$ and $\overrightarrow{BC}$. (1)
(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 $\overrightarrow{OA}$ and $\overrightarrow{OB}$. Also, find $|\vec{a}\times \vec{b}|$. (2)
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 $(\vec{a}-\vec{b})$. (2)

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 $2\hat{i}+3\hat{j}+4\hat{k}$, $7\hat{i}+5\hat{j}+8\hat{k}$ and $-3\hat{i}+7\hat{j}+11\hat{k}$ respectively.
Based on the above information, answer the following questions :
(i) How far is the star V from star A ? (1)
(ii) Find a unit vector in the direction of $\overrightarrow{DA}$. (1)
(iii) Find the measure of $\angle VDA$. (2)
OR (iii) What is the projection of vector $\overrightarrow{DV}$ on vector $\overrightarrow{DA}$ ? (2)