Let $\vec{a}$ and $\vec{b}$ be two unit vectors and $\theta$ is the angle between them. Then $\vec{a}+\vec{b}$ is a unit vector if $\theta =$

If $\overline{a}, \overline{b}, \overline{c}$ are unit vectors and $\theta$ is the angle between $\overline{a}$ and $\overline{c}$ and $\overline{a}+2 \overline{b}+2 \overline{c}=\overline{0}$,then $|\overline{a} \times \overline{c}|=$

If $|\vec{a}| = 2\sqrt{2}$,$|\vec{b}| = 3$ and the angle between $\vec{a}$ and $\vec{b}$ is $\frac{\pi}{4}$,find the length of the longer diagonal of the parallelogram whose sides are $5\vec{a} + 2\vec{b}$ and $\vec{a} - 3\vec{b}$.

If $\bar{a}, \bar{b}, \bar{c}$ are perpendicular to $\bar{b}+\bar{c}, \bar{c}+\bar{a}$ and $\bar{a}+\bar{b}$ respectively and $|\bar{a}+\bar{b}|=2, |\bar{b}+\bar{c}|=6, |\bar{c}+\bar{a}|=4$,then $|\bar{a}+\bar{b}+\bar{c}|=$

Let $\vec{a} = a_{1} \hat{i} + a_{2} \hat{j} + a_{3} \hat{k}$ with $a_{i} > 0$ for $i = 1, 2, 3$ be a vector that makes equal angles with the coordinate axes $OX$,$OY$,and $OZ$. Also,let the projection of $\vec{a}$ on the vector $3 \hat{i} + 4 \hat{j}$ be $7$. Let $\vec{b}$ be a vector obtained by rotating $\vec{a}$ by $90^{\circ}$. If $\vec{a}$,$\vec{b}$,and the $x$-axis are coplanar,then the projection of vector $\vec{b}$ on $3 \hat{i} + 4 \hat{j}$ is equal to

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