Let $|\vec{a}| = |\vec{b}| = 1$ and $|\vec{a} + \vec{b}| = \sqrt{3}$. If $\vec{c}$ is a vector satisfying the condition $\vec{c} - \vec{a} - 2\vec{b} = 3(\vec{a} \times \vec{b})$,then $\vec{c} \cdot \vec{b} = \dots$ (in $/2$)

If $|\vec{a}|=1, |\vec{b}|=2, |\vec{a}-\vec{b}|^2+|\vec{a}+2\vec{b}|^2=20$,then the angle $\theta$ between $\vec{a}$ and $\vec{b}$ is:

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.

Let $a, b, c$ be three vectors such that the magnitude of $b$ is twice that of $a$ and the magnitude of $c$ is three times that of $a$. If the angle between each pair of vectors is $\frac{\pi}{3}$ and $|a+b+c|=5$,then $|c|+|a|+|b|=$

The vector $2\hat{i} + a\hat{j} + \hat{k}$ is perpendicular to the vector $2\hat{i} - \hat{j} - \hat{k},$ if $a = $

If $l_{1}, m_{1}, n_{1}; l_{2}, m_{2}, n_{2}; l_{3}, m_{3}, n_{3}$ are the direction cosines of three mutually perpendicular lines,prove that the line whose direction cosines are proportional to $l_{1}+l_{2}+l_{3}, m_{1}+m_{2}+m_{3}, n_{1}+n_{2}+n_{3}$ makes equal angles with them.