If $a + b + c = 0$, $|a| = |b| = |c| = 3$ and $\theta$ is the angle between $b$ and $c$, then $\tan^2 \theta + \cot^2 \theta =$ (in $/3$)

If $P=(0,1,2)$, $Q=(4,-2,1)$, and $O=(0,0,0)$, then $\angle POQ$ is equal to

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

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.

If $a$ and $b$ are unit vectors and $\alpha$ is the angle between them, then $a+b$ is a unit vector when $\cos \alpha=$

Let three vectors $\overrightarrow{a}, \overrightarrow{b}$ and $\overrightarrow{c}$ be such that $\overrightarrow{c}$ is coplanar with $\overrightarrow{a}$ and $\overrightarrow{b}$,$\overrightarrow{a} \cdot \overrightarrow{c} = 7$ and $\overrightarrow{b}$ is perpendicular to $\overrightarrow{c}$,where $\overrightarrow{a} = -\hat{i} + \hat{j} + \hat{k}$ and $\overrightarrow{b} = 2\hat{i} + \hat{k}$. Then the value of $2|\overrightarrow{a} + \overrightarrow{b} + \overrightarrow{c}|^{2}$ is .........