If $|a \times b| = 4$ and $|a \cdot b| = 2$,then $|a|^2 |b|^2 = $

The cosine of the angle between any two diagonals of a cube is

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 $\theta$ is the angle between any two vectors $\vec{a}$ and $\vec{b},$ then $|\vec{a} \cdot \vec{b}| = |\vec{a} \times \vec{b}|$ when $\theta$ is equal to

$A$ unit vector in the $xy$-plane which is perpendicular to $4i - 3j + k$ is

In a quadrilateral $ABCD$,if $P$ and $Q$ are the midpoints of $\overline{BC}$ and $\overline{AD}$ respectively,then $\vec{AB} + \vec{DC} = \dots$