Statement 1: If the points (1,2,2), (2,1,2), (2,2,z) and (1,1,1) are coplanar, then z=2.

Question

Statement 1: If the points $(1,2,2)$, $(2,1,2)$, $(2,2,z)$ and $(1,1,1)$ are coplanar, then $z=2$.

Statement 2: If the 4 points $P,Q,R,S$ are coplanar, then the volume of the tetrahedron $PQRS$ is $0$.

Accepted Answer

Taking $(1,1,1)$ as base point, the vectors to the other three points are $(0,1,1)$, $(1,0,1)$, $(1,1,z-1)$. Coplanarity requires:

$$\begin{vmatrix}0&1&1\1&0&1\1&1&z-1\end{vmatrix}=0 \implies -(z-2)+1=0 \implies z=3$$

So Statement 1 (which claims $z=2$) is false. Statement 2 is a true general fact about coplanar points.

Solutions · 1