If $a$ and $b$ are the position vectors of $A$ and $B$ respectively,then the position vector of a point $C$ on $AB$ produced such that $\overrightarrow{AC} = 3\overrightarrow{AB}$ is

If the points with position vectors $10\hat{i} + 3\hat{j}$,$12\hat{i} - 5\hat{j}$,and $a\hat{i} + 11\hat{j}$ are collinear,then the value of $a$ is:

If $P$ and $Q$ are two points on the curve $y=2^{x+2}$ such that $OP \cdot \hat{i}=-1$ and $OQ \cdot \hat{i}=2$,then the magnitude of $(OQ-4OP)$ is

If $\bar{a} = \hat{i} + \hat{j} + \hat{k}$,$\bar{b} = 4\hat{i} - 2\hat{j} + 3\hat{k}$,and $\bar{c} = \hat{i} - 2\hat{j} + \hat{k}$,then the vector of magnitude $6$ units,which is parallel to the vector $2\bar{a} - \bar{b} + 3\bar{c}$,is:

If the vectors $x \hat{i}-3 \hat{j}+7 \hat{k}$ and $\hat{i}+y \hat{j}-z \hat{k}$ are collinear,then the value of $\frac{x y^2}{z}$ is equal to:

If $\vec{a}$ and $\vec{b}$ are two non-collinear vectors,then $|\vec{b}| \vec{a} + |\vec{a}| \vec{b}$ represents: