Question

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

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

Answer 1

The correct option is A $\overline{a}.\overline{b}=0,\overline{b}.\overline{c}=0,\overline{c}.\overline{a}\ne 0$

Given that,

$\begin{array}{c}\left(\overrightarrow{a}\times \overrightarrow{b}\right).\overrightarrow{c}=\left|\overrightarrow{a}\right|\left|\overrightarrow{b}\right|\left|\overrightarrow{c}\right|\\ \left|\overrightarrow{a}\times \overrightarrow{b}\right|\left|\overrightarrow{c}\right|\cos \theta =\left|\overrightarrow{a}\right|\left|\overrightarrow{b}\right|\left|\overrightarrow{c}\right|\\ \left|\overrightarrow{a}\right|\left|\overrightarrow{b}\right|\sin \theta =\left|\overrightarrow{a}\right|\left|\overrightarrow{b}\right|\\ \cos \theta =1\sin \varphi =1\\ \theta =0^0\varphi ={90}^0\\ \overrightarrow{c}=\lambda \left(\overrightarrow{a}\times \overrightarrow{b}\right)\lambda >0\\ \overrightarrow{c}.\overrightarrow{a}\ne 0\overrightarrow{c}.\overrightarrow{b}=0\overrightarrow{a}.\overrightarrow{b}=0\end{array}$

Then,

Option $A$ is correct answer.