Find the distance between the points $P(1, -3, 4)$ and $Q(-4, 1, 2)$.

Let $A, B, C$ be the feet of the perpendiculars from a point $P$ on the $xy, yz,$ and $zx$-planes respectively. Find the coordinates of $A, B, C$ for the following points $P$:
$(3, 4, 5), (-5, 3, 7), (4, -3, -5)$

Find the coordinates of a point on the $y$-axis which is at a distance of $5\sqrt{2}$ from the point $P(3, -2, 5)$.

The point $(1, -3, 4)$ lies in the octant:

Which of the following is at a distance of $\sqrt{10}$ from the point $(1, 2, 3)$?

Find the equation of the set of points $P$ such that $PA^{2} + PB^{2} = 2k^{2}$,where $A$ and $B$ are the points $(3, 4, 5)$ and $(-1, 3, -7)$,respectively.