Let $\vec{a}, \vec{b}, \vec{c}$ be three mutually perpendicular vectors of the same magnitude and equally inclined at an angle $\theta$ with the vector $\vec{a}+\vec{b}+\vec{c}$. Then $36 \cos ^{2} 2 \theta$ is equal to $.....$

Find the torque of the couple formed by forces $(9, -1, 2)$ and $(3, -2, 1)$ acting at the points $5\hat{i} + \hat{k}$ and $-5\hat{i} - \hat{k}$ respectively.

If the vectors $\bar{a}=\hat{i}-\hat{j}+2\hat{k}$,$\bar{b}=2\hat{i}+4\hat{j}+\hat{k}$ and $\bar{c}=m\hat{i}+\hat{j}+n\hat{k}$ are mutually perpendicular,then $(m, n)$ is

In a $\triangle ABC$,let $G$ denote its centroid and let $M, N$ be points in the interiors of the segments $AB, AC$,respectively,such that $M, G, N$ are collinear. If $r$ denotes the ratio of the area of $\triangle AMN$ to the area of $\triangle ABC$,then

Let $\overline{u}, \overline{v}, \overline{w}$ be vectors such that $|\overline{u}|=1, |\overline{v}|=2, |\overline{w}|=3$. If the projection of $\overline{v}$ along $\overline{u}$ is equal to that of $\overline{w}$ along $\overline{u}$,and the vectors $\overline{v}$ and $\overline{w}$ are perpendicular to each other,then $|\overline{u}-\overline{v}+\overline{w}|$ equals:

If $\bar{a}, \bar{b}, \bar{c}$ are perpendicular to $\bar{b}+\bar{c}, \bar{c}+\bar{a}$ and $\bar{a}+\bar{b}$ respectively and $|\bar{a}+\bar{b}|=2, |\bar{b}+\bar{c}|=6, |\bar{c}+\bar{a}|=4$,then $|\bar{a}+\bar{b}+\bar{c}|=$