Question

For non-zero vectors $\overline{a}$, $\overline{b}$ and $\overline{c}$, $|(\overline{a}\times \overline{b}).\overline{c}|=\left|\overline{a}\right|\left|\overline{b}\right|\left|\overline{c}\right|$ iff

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

Answer 1

The correct option is D $\overline{a}.\overline{b}=\overline{b}.\overline{c}=\overline{c}.\overline{a}=0$

$|(\overrightarrow{a}\times \overrightarrow{b}).\overrightarrow{c}|=\left|\overrightarrow{a}\right|\left|\overrightarrow{b}\right|\left|\overrightarrow{c}\right|$ $(\theta$ is angle between $\overrightarrow{c}$ and plane of $\overrightarrow{a}\times \overrightarrow{b})$

$\Rightarrow \overrightarrow{c}$ is $\perp$ to $\overrightarrow{a}$ and $\overrightarrow{b}$ for $\cos \theta =1$

and $|\overrightarrow{a}\times \overrightarrow{b}|=\left|\overrightarrow{a}\right|\left|\overrightarrow{b}\right|$ iff $\overrightarrow{a}\perp \overrightarrow{b}$

$\Rightarrow \overrightarrow{a}\cdot \overrightarrow{b}=0=\overrightarrow{b}\cdot \overrightarrow{c}=\overrightarrow{c}\cdot \overrightarrow{a}$