Question

Statement-$I$: Vector component of $\overrightarrow{b}$ perpendicular to $\overrightarrow{a}$ is $\frac{\overrightarrow{a}\times (\overrightarrow{b}\times \overrightarrow{a})}{|\overrightarrow{a}{|}^2}$

Statement-$II$: $\overrightarrow{a}\times (\overrightarrow{b}\times \overrightarrow{c}))=(\overrightarrow{a}.\overrightarrow{c})\overrightarrow{b}-(\overrightarrow{a}.\overrightarrow{b})\overrightarrow{c}$ and $|\overrightarrow{a}\times \overrightarrow{b}|=|\overrightarrow{a}||\overrightarrow{b}|\sin \theta$

• A. Both $I$ and $II$ are true and $II$ is the correct explanation of $I$
• B. Both $I$ and $II$ are true but $II$ is not correct explanation of $I$.
• C. $I$ is true, $II$ is false
• D. $I$ is false, $II$ is true

Answer 1

Answer: (B)

Both $I$ and $II$ are true but $II$ is not correct explanation of $I$.

Component of $\overline{b}$ perpendicular to $\overline{a}=\overline{b}-\frac{(\overline{a}.\overline{b})\overline{a}}{|\overline{a}{|}^2}$

Solving
$\overline{a}=\frac{(\overline{a}.\overline{a})\overline{b}-(\overline{a}.\overline{b})\overline{a}}{|\overline{a}{|}^2}$
$\overline{a}=\frac{\overline{a}\times (\overline{b}\times \overline{a})}{|\overline{a}{|}^2}$