$A$ square $ABCD$ of diagonal $2a$ is folded along the diagonal $AC$ so that the planes $DAC$ and $BAC$ are at a right angle. The shortest distance between $DC$ and $AB$ is

What is the angle between two diagonals of a cube?

Find the reflection of the point $P(2, -1, 3)$ in the plane $3x - 2y - z = 9$.

Find the image of the point $(1, 2, 3)$ in the line $\vec{r} = (6\hat{i} + 7\hat{j} + 7\hat{k}) + \lambda(3\hat{i} + 2\hat{j} - 2\hat{k})$.

The coordinates of the foot of the perpendicular drawn from point $P(1, 0, 3)$ to the line joining points $A(4, 7, 1)$ and $B(3, 5, 3)$ are:

$\Pi_1, \Pi_2, \Pi_3$ are three planes which are respectively parallel to the $YZ, ZX$ and $XY$ planes at distances $a, b$ and $c$ forming a rectangular parallelopiped. $d_1$ is a diagonal of the face of $XY$-plane not passing through the origin and $d_2$ is a diagonal of the plane $\Pi_2$ coterminous with $d_1$. If none of the coordinates of the vertices of the parallelopiped are negative, then the angle between $d_1$ and $d_2$ is