Let $A(1, 2, 3)$, $B(-1, 4, 6)$, $C(0, -6, 4)$, and $D(1, 1, 1)$ be the vertices of a tetrahedron. Let $G$ be its centroid and $G_1$ be the centroid of its face $BCD$. Then $\frac{AG_1}{AG} =$

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

$\Pi_1, \Pi_2, \Pi_3$ 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

Let $ABCD$ be a tetrahedron such that the edges $AB, AC$ and $AD$ are mutually perpendicular. Let the areas of the triangles $ABC, ACD$ and $ADB$ be $5, 6$ and $7$ square units respectively. Then the area (in square units) of the $\triangle BCD$ is equal to :

$A$ line with direction ratios $2, 1, 2$ intersects the lines $x = y + a = z$ and $x + a = 2y = 2z$. Find the coordinates of the points of intersection.

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 = $