The angle between the vectors $(2i + 6j + 3k)$ and $(12i - 4j + 3k)$ is

Let $\overline{a}, \overline{b}$ and $\overline{c}$ be vectors of magnitude $2, 3$ and $4$ respectively. If $\overline{a}$ is perpendicular to $(\overline{b}+\overline{c})$,$\overline{b}$ is perpendicular to $(\overline{c}+\overline{a})$ and $\overline{c}$ is perpendicular to $(\overline{a}+\overline{b})$,then the magnitude of $\overline{a}+\overline{b}+\overline{c}$ is equal to

$a = 4 \hat{i} + 3 \hat{j}$ 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:

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

If $\overrightarrow{a}+\overrightarrow{b}+\overrightarrow{c}=\overrightarrow{0}$,$|\overrightarrow{a}|=3$,$|\overrightarrow{b}|=5$,and $|\overrightarrow{c}|=7$,then the angle between $\overrightarrow{a}$ and $\overrightarrow{b}$ is

Let $\vec{u}$ and $\vec{v}$ be two non-zero vectors with the intermediate angle $45^{\circ}$. Then $|\vec{u} \times \vec{v}|=$