Four points are always coplanar if

Four points are always  coplanar if triple scalar  product is zero.

Step-by-step explanation:

We have four points and we find the condition for four points are coplanar.

Let A,B,C and D are four points .We have to find Vector AB, AC and AD.

Then we find triple scalar product of  vectors AB,AC and AD.

Triple scalar product=

To find triple scalar product we use determinant

Suppose A(1,0,-1),B(0,2,3),C(-2,1,1) and D(4,2,3) are four points .

We shall prove that A,B,C and D are coplanar.

We find vector AB,AC and AD

=Coordinate of B -coordinate of A=

=Coordinate of C-coordinate of A=

=Coordinate of D- coordinate of A=

Now , we find triple scalar product

Expand alon

=-1(4-4)-2(-12-6)+4(-6-3)

=0+36-36=0

Hence, triple scalar product is zero therefore, four points A,B,C and D are coplanar.

Four points are always coplanar if triple scalar product is zero.

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