Let ${\bf{\vec a}},\;{\bf{\vec b}}\;{\rm{and\;}}{\bf{\vec c}}$ be three non-coplanar vectors, and let ${\bf{\vec p}},\;{\bf{\vec q}}\;{\rm{and\;}}{\bf{\vec r}}$ be vectors defined by the relations ${\bf{\vec p}} = \frac{{{\bf{\vec b}} \times {\bf{\vec c}}}}{{\left[ {{\bf{\vec a}}\;{\bf{\vec b}}\;{\bf{\vec c}}} \right]}}{\rm{\;}},{\rm{\;}}{\bf{\vec q}} = \frac{{{\bf{\vec c}} \times {\bf{\vec a}}}}{{\left[ {{\bf{\vec a}}\;{\bf{\vec b}}\;{\bf{\vec c}}} \right]}}{\rm{and\;\;}}{\bf{\vec r}} = \frac{{{\bf{\vec a}} \times {\bf{\vec b}}}}{{\left[ {{\bf{\vec a}}\;{\bf{\vec b}}\;{\bf{\vec c}}} \right]}}{\rm{\;\;}}$ Then, the value of the expression $\left( {{\bf{\vec a}} + {\bf{\vec b}}} \right)\cdot{\bf{\vec p}} + \left( {{\bf{\vec b}} + \;{\bf{\vec c}}} \right)\cdot{\bf{\vec q}} + \left( {{\bf{\vec c}} + {\bf{\vec a}}} \right)\cdot{\bf{\vec r}}$ is equal to

JEE MATHEMATICS MCQ: SCALAR TRIPLE PRODUCT

1 Easy

1 of 1

Application +4 | -1

Let

\mathbf{\vec a},\;\mathbf{\vec b}\;\mathrm{and\;}\mathbf{\vec c}

be three non-coplanar vectors, and let

\mathbf{\vec p},\;\mathbf{\vec q}\;\mathrm{and\;}\mathbf{\vec r}

be vectors defined by
the relations

\mathbf{\vec p} = \frac{{\mathbf{\vec b} \times \mathbf{\vec c}}}{{\left[ {\mathbf{\vec a}\;\mathbf{\vec b}\;\mathbf{\vec c}} \right]}}\mathrm{\;},\mathrm{\;}\mathbf{\vec q} = \frac{{\mathbf{\vec c} \times \mathbf{\vec a}}}{{\left[ {\mathbf{\vec a}\;\mathbf{\vec b}\;\mathbf{\vec c}} \right]}}\mathrm{and\;\;}\mathbf{\vec r} = \frac{{\mathbf{\vec a} \times \mathbf{\vec b}}}{{\left[ {\mathbf{\vec a}\;\mathbf{\vec b}\;\mathbf{\vec c}} \right]}}\mathrm{\;\;}

Then, the value of the expression

\left( {\mathbf{\vec a} + \mathbf{\vec b}} \right)\cdot\mathbf{\vec p} + \left( {\mathbf{\vec b} + \;\mathbf{\vec c}} \right)\cdot\mathbf{\vec q} + \left( {\mathbf{\vec c} + \mathbf{\vec a}} \right)\cdot\mathbf{\vec r}

is equal to

View answer and explanation

Correct option: D

Option D (3) is correct because it is the result of simplifying the expression using properties of scalar triple products and reciprocal vectors.