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Question

# If $\stackrel{\to }{r}·\stackrel{\to }{a}=0,\stackrel{\to }{r}·\stackrel{\to }{b}=0\mathrm{and}\stackrel{\to }{r}·\stackrel{\to }{c}=0$ for some non-zero vector $\stackrel{\to }{r},\mathrm{then}\mathrm{the}\mathrm{value}\mathrm{of}\stackrel{\to }{a}.\left(\stackrel{\to }{b}×\stackrel{\to }{c}\right)$ is ____________.

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Solution

## Given for same non-zero vector $\stackrel{\to }{r},$ $\stackrel{\to }{r}·\stackrel{\to }{a}=0=\stackrel{\to }{r}·\stackrel{\to }{b}=\stackrel{\to }{r}·\stackrel{\to }{c}\phantom{\rule{0ex}{0ex}}\mathrm{i}.\mathrm{e}\stackrel{\to }{r}\perp \stackrel{\to }{a},\stackrel{\to }{r}\perp \stackrel{\to }{b}\mathrm{and}\stackrel{\to }{r}\perp \stackrel{\to }{c}\phantom{\rule{0ex}{0ex}}\mathrm{i}.\mathrm{e}\stackrel{\to }{a},\stackrel{\to }{b}\mathrm{and}\stackrel{\to }{c}\mathrm{are}\mathrm{coplanar}\phantom{\rule{0ex}{0ex}}\therefore \stackrel{\to }{a}·\left(\stackrel{\to }{b}×\stackrel{⇀}{c}\right)\phantom{\rule{0ex}{0ex}}=\left[\stackrel{\to }{a}\stackrel{\to }{b}\stackrel{⇀}{c}\right]\phantom{\rule{0ex}{0ex}}\stackrel{\to }{a}·\left(\stackrel{\to }{b}×\stackrel{⇀}{c}\right)=0$

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