Question

For non-zero vectors $\overrightarrow{a},\overrightarrow{b},\overrightarrow{c},(\overrightarrow{a}\times \overrightarrow{b}).\overrightarrow{c}=\left|\overrightarrow{a}\right|\left|\overrightarrow{b}\right|\left|\overrightarrow{c}\right|$ holds if and only if

• A. $\overrightarrow{a}.\overrightarrow{b}=\overrightarrow{b}.\overrightarrow{c}=0$
• B. $\overrightarrow{b}.\overrightarrow{c}=\overrightarrow{c}.\overrightarrow{a}=0$
• C. $\overrightarrow{c}.\overrightarrow{a}=\overrightarrow{a}.\overrightarrow{b}=0$
• D. $\overrightarrow{a}.\overrightarrow{b}=\overrightarrow{b}.\overrightarrow{c}=\overrightarrow{c}.\overrightarrow{a}=0$

Answer 1

Answer: (D)

$\overrightarrow{a}.\overrightarrow{b}=\overrightarrow{b}.\overrightarrow{c}=\overrightarrow{c}.\overrightarrow{a}=0$

If $\theta$ is the angle between $\overrightarrow{a}$ and $\overrightarrow{b}$and
$\varphi$ is the angle between $\overrightarrow{a}\times \overrightarrow{b}$ and $\overrightarrow{c}$, then $(\overrightarrow{a}\times \overrightarrow{b}).\overrightarrow{c}=\left|\overrightarrow{a}\right|\left|\overrightarrow{b}\right|\left|\overrightarrow{c}\right|\sin \theta \cos \varphi$
$\therefore \sin \theta =1,\cos \varphi =1$
$\Rightarrow \theta =\frac{\pi }{2},\varphi =0$
$\therefore \overrightarrow{a}\perp \overrightarrow{b},\overrightarrow{c}$ is along $\overrightarrow{a}\times \overrightarrow{b}$
$\therefore \overrightarrow{c}\perp \overrightarrow{a},\overrightarrow{c}\perp \overrightarrow{b}$
$\overrightarrow{a}.\overrightarrow{b}=\overrightarrow{b}.\overrightarrow{c}=\overrightarrow{c}.\overrightarrow{a}=0$