Question
$${\left( {\overrightarrow a \times \overrightarrow i } \right)^2} + {\left( {\overrightarrow a \times \overrightarrow j } \right)^2} + {\left( {\overrightarrow a \times \overrightarrow k } \right)^2}$$ is equal to :
A.
$${\overrightarrow a ^2}$$
B.
$$3{\overrightarrow a ^2}$$
C.
$$2{\overrightarrow a ^2}$$
D.
none of these
Answer :
$$2{\overrightarrow a ^2}$$
Solution :
\[{\left( {\overrightarrow a \times \overrightarrow i } \right)^2} = \left( {\overrightarrow a \times \overrightarrow i } \right).\left( {\overrightarrow a \times \overrightarrow i } \right) = \left| \begin{array}{l}
\overrightarrow a .\overrightarrow a \,\,\,\,\,\overrightarrow a .\overrightarrow i \\
\overrightarrow i .\overrightarrow a \,\,\,\,\,\,\overrightarrow i .\overrightarrow i
\end{array} \right| = {\overrightarrow a ^2} - {\left( {\overrightarrow a .\overrightarrow i } \right)^2}\]
$$\eqalign{
& {\text{Similarly, }}{\left( {\overrightarrow a \times \overrightarrow j } \right)^2} = {\overrightarrow a ^2} - {\left( {\overrightarrow a .\overrightarrow j } \right)^2},{\text{ and}}\,\,{\left( {\overrightarrow a \times \overrightarrow k } \right)^2} = {\overrightarrow a ^2} - {\left( {\overrightarrow a .k} \right)^2} \cr
& \therefore {\text{ the expression}} = 3{\overrightarrow a ^2} - \left\{ {{{\left( {\overrightarrow a .\overrightarrow i } \right)}^2} + {{\left( {\overrightarrow a .\overrightarrow j } \right)}^2} + {{\left( {\overrightarrow a .\overrightarrow k } \right)}^2}} \right\} \cr
& = 3{\overrightarrow a ^2} - {\overrightarrow a ^2} \cr
& = 2{\overrightarrow a ^2} \cr} $$