If $|a|=1, |b|=2$ and the angle between $a$ and $b$ is $120^{\circ}$, then ${(a+3b) \times (3a-b)}^2$ is equal to

If $a$ and $b$ are unit vectors such that $a \times b$ is also a unit vector,then the angle between $a$ and $b$ is

Let vectors $a, b, c$ and $d$ be such that $(a \times b) \times (c \times d) = 0$. If $a$ and $b$ lie in plane $P_1$ and $c$ and $d$ lie in plane $P_2$,find the angle between $P_1$ and $P_2$.

$A$ vector which is orthogonal to the vector $\vec{a} = \hat{i} + 2\hat{j} + 3\hat{k}$ and coplanar with the vectors $\vec{b} = 3\hat{i} + 2\hat{j}$ and $\vec{c} = 2\hat{i} + \hat{j} + 3\hat{k}$ is

Let $\vec{u}=2 \hat{i}-\hat{j}+\hat{k}$ and $\vec{v}=-3 \hat{j}+2 \hat{k}$ be vectors in $R^3$ and $\vec{w}$ be a unit vector in the $XY$-plane. Then,the maximum value of $|(\vec{u} \times \vec{v}) \cdot \vec{w}|$ is:

If $\vec{a} \times \vec{b} = \vec{c}$ and $\vec{b} \times \vec{c} = \vec{a}$,and $a, b, c$ are the moduli of the vectors $\vec{a}, \vec{b}, \vec{c}$ respectively,then: