Collinear Points in $3$-Dimensions

Question

The points $A(3, -1, z)$, $B(1, 2, 6)$, and $C(x, 8, 14)$ are collinear. Find the values of $x$ and $z$.

I have tried finding common ratios between the points, but no common ratio is possible, I have a feeling that this involves making the points into vectors, but I am not sure at this point?

Answer 1

I think you're assuming that the line passes through the origin.

Since $(1,2,6)$ is a point on the line, $(3,-1,z)-(1,2,6)$ and $(x,18,14)-(1,2,6)$ should have common ratios. Using this, you can figure out that $x=-3$ and $z=2$.