Let ${\bf{\vec a}},\,\,{\bf{\vec b}}\,{\rm{\;and\;}}\,\,{\bf{\vec c}}$ be three non-coplanar vectors and let ${\bf{\vec p}},\,\,{\bf{\vec q}}{\rm{\;}}\,\,\,{\rm{and}}\,{\rm{\;}}{\bf{\vec r}}$ be vector defined by the relations. ${\bf{\vec p}} = \frac{{{\bf{\vec b}} \times {\bf{\vec c}}}}{{\left[ {\;{\bf{\vec a}}\,\;{\bf{\vec b}}\,\;{\bf{\vec c}}\;} \right]}},\,\,{\bf{\vec q}} = \frac{{{\bf{\vec c}} \times {\bf{\vec a}}}}{{\left[ {{\bf{\vec a}}\;\,{\bf{\vec b}}\;\,{\bf{\vec c}}\;} \right]}}\,\,{\rm{and\;}}\,\,{\bf{\vec r}} = \frac{{{\bf{\vec a}} \times {\bf{\vec b}}}}{{\left[ {{\bf{\vec a}}\;\,{\bf{\vec b}}\;\,{\bf{\vec c}}\;} \right]}}$. 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{\;}\,\,\,\mathrm{and}\,\mathrm{\;}\mathbf{\vec r}

be vector 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]}},\,\,\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]}}

. 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 by using the properties of reciprocal vectors, we can simplify the expression to equal 3.