If $x + y + z = 0$,$|x| = |y| = |z| = 2$ and $\theta$ is the angle between $y$ and $z$,then the value of $\csc^2 \theta + \cot^2 \theta$ is equal to

Let $\vec{a}=3 \hat{i}+\hat{j}-2 \hat{k}$,$\vec{b}=4 \hat{i}+\hat{j}+7 \hat{k}$,and $\vec{c}=\hat{i}-3 \hat{j}+4 \hat{k}$ be three vectors. If a vector $\vec{p}$ satisfies $\vec{p} \times \vec{b}=\vec{c} \times \vec{b}$ and $\vec{p} \cdot \vec{a}=0$,then $\vec{p} \cdot(\hat{i}-\hat{j}-\hat{k})$ is equal to

Show that the points $A, B$ and $C$ with position vectors $\vec{a}=3 \hat{i}-4 \hat{j}-4 \hat{k}$,$\vec{b}=2 \hat{i}-\hat{j}+\hat{k}$ and $\vec{c}=\hat{i}-3 \hat{j}-5 \hat{k}$ respectively form the vertices of a right-angled triangle.

If $a, b, c$ are non-zero vectors such that $a \cdot b = a \cdot c$,then which statement is true?

If $G(\bar{g})$,$H(\bar{h})$,and $P(\bar{p})$ are respectively the centroid,orthocenter,and circumcentre of a triangle and $x \bar{p} + y \bar{h} + z \bar{g} = \overline{0}$,then $x, y, z$ are respectively:

If the lines $\vec{r} = 2\hat{i} + \hat{j} + \hat{k} + \lambda(\hat{i} - 2\hat{j})$ and $\vec{r} = \hat{i} + \hat{j} - 3\hat{k} + \mu(\hat{j} + 2\hat{k})$ intersect each other,then $(\lambda + \mu)$ is equal to