Question

If [a b c]=[b c d]+[c a d]+[a b d] then show that the points with position vectors a,b,c and d are coplanar.

Answer 1

Answer:

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]}Read more on Sarthaks.com - https://www.sarthaks.com/490080/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