Coplanarity of 4 points with one of the points having a variable Z coordinate

Question

There 4 points,

$A \equiv (2, 1, -3)$

$B \equiv (3, -2, 1)$

$C \equiv (-1, 3, 2)$

$D \equiv (-3, -3, q)$

My question is how many values can $q$ take such that these 4 coordinates are coplanar?

I observed that there are more than one. I wish to know if I have made a mistake and if I haven't, I'd like know to how many values $q$ can take so $A, B, C$ and $D$ are coplanar.

Answer 1

Since $\vec{AB} = (1,-3,4)$ and $\vec{AC}= (-3,2,5)$ we have $\vec{AB}\ne k\vec{AC}$ for each real $k$ so $A,B,C$ are not colinear, so they determine plane.

So I would say exactly one $q$. Since $\vec{AD}$ can be expressed in unique way with $\vec{AB}$ and $\vec{AC}$ if $D$ lies on plane $ABC$.