If $|\bar{a} \times \bar{b}|^2+(\bar{a} \cdot \bar{b})^2=144$ and $|\bar{a}|=4$,then $|\bar{b}|=$

If $\overline{a} = 2 \hat{i} + 3 \hat{j} - 4 \hat{k}$ and $\overline{b} = \hat{i} - \hat{j} - \hat{k}$,then the projection of $\overline{b}$ in the direction of $\overline{a}$ is

If the position vectors of $A$ and $B$ are $6i + j - 3k$ and $4i - 3j - 2k$ respectively,then the work done by the force $\vec{F} = i - 3j + 5k$ in displacing a particle from $A$ to $B$ is ............ $units$.

If $3 \hat{j}$,$4 \hat{k}$ and $3 \hat{j}+4 \hat{k}$ are the position vectors of the vertices $A$,$B$,and $C$ respectively of $\triangle ABC$,then the position vector of the point in which the bisector of $\angle A$ meets $BC$ is

The position vectors of the vertices of a triangle $ABC$ are $4i - 2j$,$i + 4j - 3k$,and $-i + 5j + k$ respectively. Then $\angle ABC = $

The angle between the vectors $a = 2i + 3j + k$ and $b = 2i - j - k$ is