For non-zero vectors →a,→b,→c;∣∣∣(→a×→b)⋅→c∣∣∣=∣∣→a∣∣∣∣∣→b∣∣∣∣∣→c∣∣ holds if and only if
A
→a⋅→b=0,→b⋅→c=0
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B
→b⋅→c=0,→c⋅→a=0
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C
→c⋅→a=0,→a⋅→b=0
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D
→a⋅→b=→b⋅→c=→c⋅→a=0
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Solution
The correct option is C→a⋅→b=→b⋅→c=→c⋅→a=0 We 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.