If $\left(\frac{9}{4}, \frac{5}{4}, \frac{15}{4}\right)$ is the centroid of a tetrahedron whose vertices are $(a, 2, 1), (1, b, 4), (4, 0, c)$ and $(1, 1, 7)$, then

Let $A(3, 0, -1)$,$B(2, 10, 6)$,and $C(1, 2, 1)$ be the vertices of a triangle and $M$ be the midpoint of $AC$. If $G$ divides $BM$ in the ratio $2 : 1$,then $\cos(\angle GOA)$ ($O$ being the origin) is equal to

Find the reflection of the point $(1, 6, 3)$ in the line $\frac{x}{1} = \frac{y - 1}{2} = \frac{z - 2}{3}$.

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

$A(1,1,1), B(1,-4,3), C(2,-2,0)$ and $D(8,1,4)$ are the vertices of a tetrahedron. $G_1, G_2, G_3$ and $G_4$ are the centroids of the faces $ABC, BCD, CDA$ and $DAB$. Then the centroid of the tetrahedron having $G_1, G_2, G_3, G_4$ as its vertices is

Let three points $A(2,3,4), B(3,4,2)$ and $C(4,2,3)$ in space be given. $A$ point $D$ in space is such that it is at a distance of $\sqrt{6}$ units from the $3$ given points. Then the volume of the tetrahedron $ABCD$ is -