The length of the internal bisector of angle $A$ in $\triangle ABC$ with vertices $A(4,7,8)$,$B(2,3,4)$ and $C(2,5,7)$ is

If $P \equiv (0, 1, 0)$ and $Q \equiv (0, 0, 1)$,then the projection of $PQ$ on the plane $x + y + z = 3$ is

Let $ABCD$ be a tetrahedron in which the coordinates of each of its vertices are in arithmetic progression with the same common difference. If the centroid $G$ of the tetrahedron is $(2, 3, k)$,then the distance of $G$ from the origin is

If a line makes angles $\alpha, \beta, \gamma, \delta$ with the four diagonals of a cube,then the value of $\cos^2 \alpha + \cos^2 \beta + \cos^2 \gamma + \cos^2 \delta = $

If the ratio of the perpendicular distances of a variable point $P(x, y, z)$ from the $X$-axis and from the $YZ$-plane is $2:3$, then the equation of the locus of $P$ is

$A$ 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