if vectors a,b and c are coplanar then prove that vector a+b,b+c,c+a are coplanar - Maths - Vector Algebra

Question

if vectors a,b and c are coplanar then prove that vector
a+b,b+c,c+a are coplanar

Gursheen kaur. · Accepted Answer

If a, b and c are coplanar vectors Their scalar triple product will be zero i e $[a \cdot (b \times c)] = 0$ Now, Scalar triple product of given vectors, $[(a + b) \cdot ((b + c) \times (c + a))] = [a \cdot ((b + c) \times (c + a))] + [b \cdot ((b + c) \times (c + a))] = [a \cdot ((b + c) \times c + (b + c) \times a)] + [b \cdot ((b \times (c + a) + c \times (c + a))] = [a \cdot ((b + c) \times c)] + 0 + 0 + [b (c \times (c + a))] = [a \cdot (b \times c + c \times c)] + [b \cdot (c \times c + c \times a)] = [a \cdot (b \times c)] + 0 + 0 + [b \cdot (c \times a)] = [a (b \times c)] + [a \cdot (b \times c)] = 0 + 0 = 0$ Thus, the given vectors are coplanar