If $\vec{u}$ and $\vec{v}$ are unit vectors and $\theta$ is the acute angle between them,then $2\vec{u} \times 3\vec{v}$ is a unit vector for

Let the vectors $\overline{a}, \overline{b}, \overline{c}$ and $\overline{d}$ be such that $(\overline{a} \times \overline{b}) \times(\overline{c} \times \overline{d})=\overline{0}$. Let $P_1$ and $P_2$ be the planes determined by the pair of vectors $\overline{a}, \overline{b}$ and $\overline{c}, \overline{d}$ respectively,then the angle between $P_1$ and $P_2$ is

If non-zero vectors $\vec{a}$ and $\vec{b}$ are perpendicular to each other,then the solution of the equation $\vec{r} \times \vec{a} = \vec{b}$ is:

Find the area of a triangle having the points $A(1, 1, 1)$,$B(1, 2, 3)$,and $C(2, 3, 1)$ as its vertices.

Let $\vec{p}=2 \hat{i}+3 \hat{j}+\hat{k}$ and $\vec{q}=\hat{i}+2 \hat{j}+\hat{k}$ be two vectors. If a vector $\vec{r}=(\alpha \hat{i}+\beta \hat{j}+\gamma \hat{k})$ is perpendicular to each of the vectors $(\vec{p}+\vec{q})$ and $(\vec{p}-\vec{q})$,and $|\vec{r}|=\sqrt{3}$,then $|\alpha|+|\beta|+|\gamma|$ is equal to $.....$

If $a, b$ and $c$ are position vectors of the vertices of $\triangle ABC$,then $\frac{(a-c) \times (b-a)}{(b-a) \cdot (c-a)} = $