The centroid of a tetrahedron with vertices $P(5, -7, 0)$,$Q(a, 5, 3)$,$R(4, -6, b)$,and $S(6, c, 2)$ is $(4, -3, 2)$. Then the value of $2a + 3b + c$ is equal to:

The incenter of the triangle $ABC$,whose vertices are $A(0,2,1)$,$B(-2,0,0)$,and $C(-2,0,2)$ is

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

Find the length of the perpendicular drawn from the point $P(2\hat{i} - \hat{j} + 5\hat{k})$ to the line $\vec{r} = (11\hat{i} - 2\hat{j} - 8\hat{k}) + \lambda(10\hat{i} - 4\hat{j} - 11\hat{k})$.

Let $A(1, 8, 4)$, $B(0, -11, 3)$, and $C(2, -3, -1)$ be three points. If $D$ is the foot of the perpendicular from $A$ to the line $BC$, find the coordinates of $D$.

Find the point in the $XY$-plane which is equidistant from the three points $A(2, 0, 3)$,$B(0, 3, 2)$,and $C(0, 0, 1)$.