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  4. For non-zero vectors a,b,c, |(a×b).c|=| a || b || c | holds, if and only if

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

IIT JEEIIT JEE 1982Vector Algebra

Solution:

Given, $|(\overrightarrow{ a } \times \overrightarrow{ b }) \cdot \overrightarrow{ c }|=|\overrightarrow{ a }||\overrightarrow{ b }||\overrightarrow{ c }|$
$\Rightarrow\|\overrightarrow{ a }\| \overrightarrow{ b }|\sin \theta \hat{ n } \cdot \overrightarrow{ c }|=|\overrightarrow{ a } \| \overrightarrow{ b }||\overrightarrow{ c }|$
$\Rightarrow |\overrightarrow{ a }||\overrightarrow{ b }||\overrightarrow{ c }||\sin \theta \cdot \cos \alpha|=|\overrightarrow{ a }||\overrightarrow{ b }||\overrightarrow{ c }|$
$\Rightarrow |\sin \theta| \cdot|\cos \alpha|=1 \Rightarrow \theta=\frac{\pi}{2} $ and $ \alpha=0$
$\therefore \overrightarrow{ a } \perp \overrightarrow{ b }$ and $\overrightarrow{ c } \| \hat{ n }$
i.e. $\overrightarrow{ a } \perp \overrightarrow{ b }$ and $\overrightarrow{ c }$ perpendicular to both $\overrightarrow{ a }$ and $\overrightarrow{ b }$