JEE Main & Advanced Mathematics Vector Algebra Vector Triple Product

Let \[\mathbf{a},\,\mathbf{b},\,\mathbf{c}\] be any three vectors, then the vectors \[\mathbf{a}\times (\mathbf{b}\times \mathbf{c})\] and \[(\mathbf{a}\times \mathbf{b})\,\,\times \mathbf{c}\] are called vector triple product of \[\mathbf{a,b,}\,\mathbf{c}\] .

Thus, \[\mathbf{a}\,\times \,(\mathbf{b}\times \mathbf{c})=(\mathbf{a}\,.\,\mathbf{c})\,\mathbf{b}-(\mathbf{a}\,.\,\mathbf{b})\mathbf{c}\]

Properties of vector triple product

(i) The vector triple product \[\mathbf{a}\times (\mathbf{b}\,\times \,\mathbf{c})\] is a linear combination of those two vectors which are within brackets.

(ii) The vector \[\mathbf{r}=\mathbf{a}\times (\mathbf{b}\times \mathbf{c})\] is perpendicular to \[\mathbf{a}\] and lies in the plane of \[\mathbf{b}\] and \[\mathbf{c}\].

(iii) The formula \[\mathbf{r}={{\mathbf{a}}_{1}}+\lambda \mathbf{b}\] is true only when the vector outside the bracket is on the left most side. If it is not, we first shift on left by using the properties of cross product and then apply the same formula.

Thus, \[\mathbf{(b\times c)\times a=}-\mathbf{\{a\times (b\times c)\}=}-\mathbf{\{(a}\mathbf{.c)b}-\mathbf{(a}\mathbf{.b)c\}=(a}\mathbf{.b)c}-\mathbf{(a}\mathbf{.c)b}\]

(iv) Vector triple product is a vector quantity.

(v) \[\mathbf{a}\times (\mathbf{b}\times \mathbf{c})\ne (\mathbf{a}\times \mathbf{b})\times \mathbf{c}\]