Question

To prove the result \(a \times (b \times c) + b \times (c \times a) + c \times (a \times b) = 0\).

Short answer

The result is proved.

Step-by-step solution

Given data

\(a \times (b \times c) + b \times (c \times a) + c \times (a \times b) = 0\)

Concept used of cross product

If\(\theta \)is the angle between the given two vectors\(A\)and\(B\), then the formula for the cross product of vectors is given by:

\(A \times B = AB\sin \theta \).

Prove the result

The vector \(b \times (c \times a)\) is equivalent to \((b \cdot a)c - (b \cdot c)a\). That is, \(b \times (c \times a) = (b \cdot a)c - (b \cdot c)a\).

Similarly, the equivalent vector \(c \times (a \times b)\) is \((c \cdot b)a - (c \cdot a)b\). That is, \(c \times (a \times b) = (c \cdot b)a - (c \cdot a)b\)

Consider the left hand side of the result \(a \times (b \times c) + b \times (c \times a) + c \times (a \times b) = 0\).

L. H.S

Hence, the left and right hand side of the given result is true.