= [ , , ] [ , , ] [ , , ] b c a b c a a b c . Similarly, the remaining equalities are proved. We have studied about coplanar vectors in XI standard as three nonzero vectors of which, one can be expressed as a linear combination of the other two. Now we use scalar triple product for the characterisation of coplanar vectors.
Theorem . The scalar triple product of three non-zero vectors is zero if, and only if, the three vectors are coplanar. Proof Let , , a b c be any three non-zero vectors. Then, a b = ⇔ c is perpendicular to a b ⇔ c lies in the plane which is parallel to both a and b ⇔ , , a b c are coplanar.
Theorem . Three vectors , , a b c are coplanar if, and only if, there exist scalars , , r s t ∈ such that atleast one of them is non-zero and ra sb tc Vector - - Applications of Vector Algebra Proof Let a i a j a k b b i b j b k c c i c j c k . Then, we have , , a b c are coplanar ⇔ , , a b c = ⇔ = ⇔ there exist scalars , , r s t ∈ , atleast one of them non-zero such that a r a s a t = , b r b s b t = , c r c s c t = ⇔ there exist scalars , , r s t ∈ , atleast one of them non-zero such that ra sb tc Theorem . If , , a b c and , , p q r are any two systems of three vectors, and if p , x a y b z c q , x a y