Question
If $$\overrightarrow a ||\overrightarrow b \times \overrightarrow c $$ then $$\left( {\overrightarrow a \times \overrightarrow b } \right).\left( {\overrightarrow a \times \overrightarrow c } \right)$$ is equal to :
A.
$${\overrightarrow a ^2}\left( {\overrightarrow b .\overrightarrow c } \right)$$
B.
$${\overrightarrow b ^2}\left( {\overrightarrow a .\overrightarrow c } \right)$$
C.
$${\overrightarrow c ^2}\left( {\overrightarrow a .\overrightarrow b } \right)$$
D.
none of these
Answer :
$${\overrightarrow a ^2}\left( {\overrightarrow b .\overrightarrow c } \right)$$
Solution :
$$\overrightarrow a ||\left( {\overrightarrow b \times \overrightarrow c } \right)\,\,\, \Rightarrow \overrightarrow a \times \left( {\overrightarrow b \times \overrightarrow c } \right) = 0\,\,\, \Rightarrow \left( {\overrightarrow a .\overrightarrow c } \right)\overrightarrow b = \left( {\overrightarrow a .\overrightarrow b } \right)\overrightarrow c $$
But $${\overrightarrow b }$$ is not collinear with $${\overrightarrow c }.$$ So, $$\overrightarrow a .\overrightarrow c = 0{\text{ and }}\overrightarrow a .\overrightarrow b = 0$$
Now, \[\left( {\overrightarrow a \times \overrightarrow b } \right).\left( {\overrightarrow a \times \overrightarrow c } \right) = \left| \begin{array}{l}
\overrightarrow a .\overrightarrow a \,\,\,\,\,\overrightarrow a .\overrightarrow c \\
\overrightarrow b .\overrightarrow a \,\,\,\,\,\overrightarrow b .\overrightarrow c
\end{array} \right| = \left| \begin{array}{l}
\overrightarrow a .\overrightarrow a \,\,\,\,\,0\\
0\,\,\,\,\,\,\overrightarrow b .\overrightarrow c
\end{array} \right| = {\overrightarrow a ^2}\left( {\overrightarrow b .\overrightarrow c } \right).\]