Let $\vec{a}, \vec{b}$ and $\vec{c}$ be three unit vectors such that $|\vec{a}-\vec{b}|^{2}+|\vec{a}-\vec{c}|^{2}=8$. Then $|\vec{a}+2\vec{b}|^{2}+|\vec{a}+2\vec{c}|^{2}$ is equal to

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

If the position vectors of the vertices $A, B$ and $C$ of $\triangle ABC$ are $\hat{i}+2\hat{j}-5\hat{k}$,$-2\hat{i}+2\hat{j}+\hat{k}$ and $2\hat{i}+\hat{j}-\hat{k}$ respectively,then $\angle B=$

Let the position vectors of the points $P, Q, R$ and $S$ be $\vec{a}=\hat{i}+2 \hat{j}-5 \hat{k}$,$\vec{b}=3 \hat{i}+6 \hat{j}+3 \hat{k}$,$\vec{c}=\frac{17}{5} \hat{i}+\frac{16}{5} \hat{j}+7 \hat{k}$ and $\vec{d}=2 \hat{i}+\hat{j}+\hat{k}$,respectively. Then which of the following statements is true?

If $4\hat{i}+\hat{j}-\hat{k}$ and $3\hat{i}+m\hat{j}+2\hat{k}$ are at a right angle,then $m = $

If the line joining points $A$ and $B$ having position vectors $6 \vec{a}-4 \vec{b}+4 \vec{c}$ and $-4 \vec{c}$ respectively,and the line joining the points $C$ and $D$ having position vectors $-\vec{a}-2 \vec{b}-3 \vec{c}$ and $\vec{a}+2 \vec{b}-5 \vec{c}$ intersect,then their point of intersection is