For the problems below assume all vectors A B C D are three

For the problems below, assume all vectors A, B, C, D are three dimensions. Let C = A - B, calculate the dot product of C. Is the following true: (A times B) times C = A times (B times C)

Solution

^(a)Let A = ia₁+ja₂+ka₃ , B = ib₁+jb₂+kb₃ , C = ic₁+jc₂+kc₃ and D = id₁+jd₂+kd₃. Then C = A –B = i(a₁– b₁) +j(a₂- b₂)+k(a₃–b₃) and hence C xC = (A-B)x(A-B) = (a₁– b₁)² + (a₂- b₂)²+ (a₃–b₃)² = square of the magnitude of C.

^(b)The dot product of vectors is not associative i.e. (AxB) x C A x(BxC), since the dot product of 2 vectors is a scalar and the dot product of a scalar with a vector is not defined. Here AxB is a scalar and (AxB) x C is not defined. Similarly, A x(BxC) is not defined as BxC is a scalar.