Prove that the necessary and sufficient condition for any four points in three-dimensional space to be coplanar is that there exists a liner relation connecting their position vectors such that the algebraic sum of the coefficients (not all zero) in it is zero.

Explanation

Let us assume that the points `A, B, C and D` whose position vectors are `veca, vecb, vecc and vecd`, respectively, are coplaner. In the case the lines AB and CD will intersect at some point P (it being assumed that AB and CD are not parallel, and if they are, then we will choose any other pair of non-parallel lines formed by the given points ). If P divides AB in the ratio `q:p` and CD in the ratio `n:m` , then the position vector of P written from AB and CD is
`" "(pveca+qvecb)/(p+q)=(mvecc+nvecd)/(m+n)`
or `" "(p)/(p+q)veca+(q)/(p+q)vecb-(m)/(m+n)vecc-(n)/(m+n)vecd=vec0`
or `" "Lveca+Mvecb+Nvecc+Pvecd=vec0`
where `" "L+M+N+P= (p)/(p+q)+(q)/(p+q)-(m)/(m+n)-(n)/(m+n)=1-1=0`
Hence, the condition is necessary.