Question

A vector $\overrightarrow{R}$ is given by $\overrightarrow{R}=\overrightarrow{A}\times (\overrightarrow{B}\times \overrightarrow{C}).$ Which of the following is true?

• A. $\overrightarrow{R}$ is parallel to $\overrightarrow{A}$
• B. $\overrightarrow{R}$ must be parallel to $\overrightarrow{B}$
• C. $\overrightarrow{R}$ must be perpendicular to $\overrightarrow{B}$
• D. None of the options

Answer 1

The correct option is D None of the options

Let $\overrightarrow{A}=\hat{i},\overrightarrow{B}=\hat{i}+\hat{j},\overrightarrow{C}=\hat{k}$
$\overrightarrow{R}=\overrightarrow{A}\times (\overrightarrow{B}\times \overrightarrow{C})=\hat{i}\times \left[\right(\hat{i}+\hat{j})\times \hat{k}]=\hat{i}\times [-\hat{j}+\hat{i}]=-\hat{k}$
$\overrightarrow{R}$ must be perpendicular to both $\overrightarrow{A}$ and $(\overrightarrow{B}\times \overrightarrow{C})$ according to the given equation, but we got $\overrightarrow{R}$ is neither parallel nor perpendicular to $\overrightarrow{B}$.