Question

The vector $\overrightarrow{a}\times (\overrightarrow{b}\times \overrightarrow{c})$ is

• A. Perpendicular to $\overrightarrow{b}$ and parallel to $\overrightarrow{c}$
• B. Coplanar with $\overrightarrow{b}and\overrightarrow{c}$ and orthogonal to $\overrightarrow{a}$
• C. (c) Perpendicular to both $\overrightarrow{b}and\overrightarrow{c}$
• D. None of the above

Answer 1

The correct option is B Coplanar with $\overrightarrow{b}and\overrightarrow{c}$ and orthogonal to $\overrightarrow{a}$

Look its simple.
Let $\overrightarrow{u}=\overrightarrow{a}\times (\overrightarrow{b}\times \overrightarrow{c})$
­­­From the definition of cross product $\overrightarrow{u}$ is perpendicular to $\overrightarrow{a}and\overrightarrow{b}\times \overrightarrow{c}$ …. (1)
Now $\overrightarrow{u}$ is perpendicular to $\overrightarrow{b}\times \overrightarrow{c}$.

See the figure. If b and c are the vectors (mentioned in blue) in the plane as shown then $b\times c$ will be perpendicular to that plane as shown by orange arrow. This orange arrow is in the direction of $b\times c$
Now from statement 1 we can say that
$\overrightarrow{u}$ is perpendicular to the plane of $b\times c$. The plane perpendicular to the plane of $b\times c$ (the orange arrow) is the plane which contains $\overrightarrow{b}and\overrightarrow{c}$
This implies $\overrightarrow{u}$ lies in the plane of $\overrightarrow{b}and\overrightarrow{c}$.
Also from the statement (1), $\overrightarrow{u}$ is orthogonal to $\overrightarrow{a}$ (Definition of cross product).
So $\overrightarrow{u}$ is coplanar with $\overrightarrow{b}and\overrightarrow{c}$ and orthogonal to $\overrightarrow{a}$.