# 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 $$\left| \begin{array}{ccc} b_1 & b_2 & b_3 \\ c_1 & c_2 & c_3 \\ d_1 & d_2 & d_3 \end{array} \right|=0$$

• Thanks. Can you tell me and one questions: how to find a dictance for the given two line l_1: 13x-y+6z+6=0; l_2: 26x-2y+12z-4=0 – he hehi Dec 2 '15 at 21:40

Take any three of the points and determine the equation of the plane. As TonyK said, three points always belong to one plane and, if they do not all lie in a line, then the determine a unique plane. Once you have the equation of the plane, put the coordinates of the fourth point into the equation to see if it is satisfied. If the three points you chose do happen to lie on a single line then you are done- any fourth point will determine a plane that all four points lie on.

• For example : A(0,2,4); B((5,1,2); C(3,8,3); D(2,-2,1) how to proced – he hehi Dec 2 '15 at 21:30
• @hehehi In your example, points A, B, and C determine a plane. Find the equation of that plane. Do the coordinates of D satisfy that equation? – David K Dec 2 '15 at 21:55