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 $/3$)

Show that the vectors $2 \hat{i}-\hat{j}+\hat{k}, \hat{i}-3 \hat{j}-5 \hat{k}$ and $3 \hat{i}-4 \hat{j}-4 \hat{k}$ form the vertices of a right-angled triangle.

Let $a, b$ and $c$ be unit vectors such that $a$ is perpendicular to the plane containing $b$ and $c$ and the angle between $b$ and $c$ is $\frac{\pi}{3}$. Then,$|a+b+c|=$

If $|\vec{a}|=13, |\vec{b}|=5$ and $\vec{a} \cdot \vec{b}=60$,then $|\vec{a} \times \vec{b}|=$

If $\theta$ is the angle between two vectors $\vec{a}$ and $\vec{b},$ then $\vec{a} \cdot \vec{b} \geq 0$ only when

$a = 4 \hat{i} + 3 \hat{j}$ and $b$ are two vectors in the $XOY$ plane, and $a$ is perpendicular to $b$. $A$ vector $c$ lying in the same plane and having projections $1$ and $2$ on $a$ and $b$ respectively is: