# How to prove four points belong to the same plane

How can he prove the points $A=(a_1, a_2, a_3)$, $B=(b_1, b_2, b_3)$, $C=(c_1, c_2, c_3)$, $D=(d_1, d_2, d_3)$ belong to the same plane, as if they belong can then find plane. I know how to prove three given points belong the same plane.

• Three given points always belong to the same plane. – TonyK Dec 2 '15 at 21:20
• Do you know determinants? – Wojowu Dec 2 '15 at 21:25
• Yes. I now the determinants – he hehi Dec 2 '15 at 21:26

Hint: Translate by $-A$ so that we can assume $A=(0,0,0)$. Then points $A,B,C,D$ lie on a plane iff vectors $B,C,D$ do not span whole space, i.e. if $$\left| \begin{array}{ccc} b_1 & b_2 & b_3 \\ c_1 & c_2 & c_3 \\ d_1 & d_2 & d_3 \end{array} \right|=0$$