Q:

Proof of test of collinearity and coplanarity

Statement : If there are 3 points with position vectors a, b and c. Then the points are collinear if and only if there exist scalars x,y,z, not all zero,such that x a + y b +z c = 0 where x+y+z =0.
Statement : If there are 4 points with position vectors a, b ,c and d. Then the points are copalanar if and only if there exist scalars x,y,z and w not all zero,such that x a + y b +z c +w d = 0 where x+y+z+w =0.
Can you give a proof for this using vectors?

A:

Since $x+y+z = 0$ at least one of $x, y$ is not zero and : $$x\mathbf a + y\mathbf b + z\mathbf c = 0 \Leftrightarrow x(\mathbf a - \mathbf c) + y(\mathbf b - \mathbf c) = 0.$$ Two dependent vectors lie on a single line through the origin. This works the other way around as well.
Extend 1. to cover the coplanar case.

