Question

If $a$ is perpendicular to both $b$ and $c$, then?

• A. $a+(b+c)=0$
• B. $a\times (b+c)=0$
• C. $a\times (b\times c)=0$
• D. $a·(b\times c)=0$

Answer 1

The correct option is C

$a\times (b\times c)=0$

Explanation for the correct option:

Given, $\overrightarrow{a}\perp \overrightarrow{b},\overrightarrow{a}\perp \overrightarrow{c}$

As we know,

$\overrightarrow{a}\times (\overrightarrow{b}\times \overrightarrow{c})=(\overrightarrow{a}\cdot \overrightarrow{c})\overrightarrow{b}-(\overrightarrow{a}\cdot \overrightarrow{b})\overrightarrow{c}$

$=0-0$ $\left[\because \overrightarrow{a}\perp \overrightarrow{b},\overrightarrow{a}\perp \overrightarrow{c},\therefore \overrightarrow{a}·\overrightarrow{b}=0,\overrightarrow{a}·\overrightarrow{c}=0\right]$

$\mathbf{\therefore }\overrightarrow{\mathbf{a}}\mathbf{}\mathbf{\times }\mathbf{}\left(\overrightarrow{b}\times \overrightarrow{c}\right)\mathbf{}\mathbf{=}\mathbf{}\mathbf{0}$

Hence, Option ‘C’ is Correct.