if a- ,b- , c- are coplanar vectors prove that a+b , b+c ,c+a are also coplanar - Maths - Vector Algebra

Question

if a->,b->, c-> are
coplanar vectors prove that a+b , b+c ,c+a are also coplanar

Ajanta Trivedi · 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 we will find scalar triple product of given vectors, $[(a + b) \cdot ((b + c) \times (c + a))] = [a \cdot ((b + c) \times (c + a))] + [b \cdot ((b + c) \times (c + a))] = [a \cdot ((b + c) \times c + (b + c) \times a)] + [b \cdot ((b \times (c + a) + c \times (c + a))] = [a \cdot ((b + c) \times c)] + 0 + 0 + [b (c \times (c + a))] = [a \cdot (b \times c + c \times c)] + [b \cdot (c \times c + c \times a)] = [a \cdot (b \times c)] + 0 + 0 + [b \cdot (c \times a)] = [a (b \times c)] + [a \cdot (b \times c)] = 0 + 0 = 0$ hence the given vectors are coplanar

if a->,b->, c-> are coplanar vectors prove that a+b , b+c ,c+a are also coplanar

if a^->,b^->, c^-> are coplanar vectors prove that a+b , b+c ,c+a are also coplanar