In $\triangle OAC$,if $B$ is the mid-point of side $AC$ and $\vec{OA}=\vec{a}, \vec{OB}=\vec{b}$,then $\vec{OC}$ is equal to

$A$ vector which is in the direction of $(3, 6, 2)$ and has magnitude $4$ is $.......$

If $P \hat{i}-2 \hat{j}+3 \hat{k}$, $2 \hat{i}+3 \hat{j}-4 \hat{k}$, and $4 \hat{i}+13 \hat{j}-18 \hat{k}$ are the position vectors of three collinear points $A$, $B$, and $C$ respectively, then the vector in the direction of $AB$ of length $|P|$ units is

In a triangle $ABC$,if $2\overrightarrow{AC} = 3\overrightarrow{CB}$,then $2\overrightarrow{OA} + 3\overrightarrow{OB}$ equals

If the position vectors of the points $A, B, C, D$ are $\vec{a} = \hat{i} + \hat{j} + \hat{k}$,$\vec{b} = 2\hat{i} + 5\hat{j}$,$\vec{c} = 3\hat{i} + 2\hat{j} - 3\hat{k}$,and $\vec{d} = \hat{i} - 6\hat{j} - \hat{k}$,then the angle between the vectors $\overrightarrow{AB}$ and $\overrightarrow{CD}$ is:

If $C$ is the midpoint of $AB$ and $P$ is any point outside $AB$,then