Show that the four points `A ,B , Ca n dD` with position vectors ` vec a , vec b , vec c` and ` vec d` respectively are coplanar if and only if `3 vec a-2 vec b+ vec c-2 vec d=0.`

Explanation

We are given that


`A,B,C,D` has the position vectors `vec a, vec b, vec c, vec d`


`3 vec a-2 vec b+ vec c-2 vec d=0`


`rArr 3 vec a+vec c=2 vec b+2 vec d`


`rArr 3 vec(OA)+ vec(OC)=2 vec(OB)+2 vec(OD)`


This shows that there is a linear combination of the two vectors. Thus the vectors are coplanar.