Question

Show that the four points vectors a, b, c, d are coplanar iff [b c d] + [c a d] + [a b d] = [a b c]

Answer 1

Let A, B, C, D be the given points respectively

A, B, C, D are coplanar 

⇔ vector[AB AC AD] = 0

⇔ vector[OB - OA OC - OA OD - OA] = 0

⇔ vector[b-a c-a d-a]= 0

⇔ vector[b c -a d-a] - vector[a c-a d-a] = 0

⇔ vector{[b c d-a] - [b a d-a] - [a c d-a] + [a a d-a]} = 0

⇔ vector{[b c d] - [b c a] - [b a d] + [b a a] - [a c d]} + 0 = 0

⇔ vector{[b c d] - [a b c] + [a b d] + 0 +[c a d]} + 0 = 0

⇔ vector{[b c d] + [a b d] + [c a d] = [a b c]}