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Question

# For non-zero vectors →a,→b,→c;∣∣∣(→a×→b)⋅→c∣∣∣=∣∣→a∣∣∣∣∣→b∣∣∣∣∣→c∣∣ holds if and only if

A
ab=0,bc=0
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B
bc=0,ca=0
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C
ca=0,ab=0
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D
ab=bc=ca=0
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Solution

## The correct option is C →a⋅→b=→b⋅→c=→c⋅→a=0We have ∣∣∣(→a×→b)⋅→c∣∣∣=∣∣→a∣∣∣∣∣→b∣∣∣∣∣→c∣∣⇔∣∣∣∣∣→a∣∣∣∣∣→b∣∣∣sinθ^n⋅→c∣∣∣=∣∣→a∣∣∣∣∣→b∣∣∣∣∣→c∣∣⇔∣∣∣∣∣→a∣∣∣∣∣→b∣∣∣∣∣→c∣∣sinθcosα∣∣∣=∣∣→a∣∣∣∣∣→b∣∣∣∣∣→c∣∣⇔|sinθ||cosα|=1⇔θ=π2 and α=0⇔a⊥→b and →c is parallel to ^n⇔a⊥→b and →c⊥ to both →a and →b⇔ a, b, c are mutually perpendicular ⇔→a⋅→b=→b⋅→c=→c⋅→a=0 Hence (D) is correct.

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