Let a→,b→and c→ be three unit vectors such that a→+b→+c→=0→. If λ=a→.b→+b→.c→+c→.a→and then d→=a→×b→+b→×c→+c→×a→the ordered pair,λ,d→ is equal to:

a→+b→+c→2=03+2a→.b→+b→.c→+c→.a→=0a→.b→+b→.c→+c→.a→=−32⇒λ=−32d→=a→×b→+b→×−a→−b→+−a→−b→×a→=a→×b→+a→×b→+a→×b→d→=3a→×b→

Let a→,b→and c→ be three unit vectors such that a→+b→+c→=0→. If λ=a→.b→+b→.c→+c→.a→and then d→=a→×b→+b→×c→+c→×a→the ordered pair,λ,d→ is equal to:

a→+b→+c→2=03+2a→.b→+b→.c→+c→.a→=0a→.b→+b→.c→+c→.a→=−32⇒λ=−32d→=a→×b→+b→×−a→−b→+−a→−b→×a→=a→×b→+a→×b→+a→×b→d→=3a→×b→

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Let $\vec{a}, \vec{b}$ and $\vec{c}$ be three unit vectors such that $\vec{a} + \vec{b} + \vec{c} = \vec{0}$ . If $λ = \vec{a} . \vec{b} + \vec{b} . \vec{c} + \vec{c} . \vec{a}$ and then $\vec{d} = \vec{a} \times \vec{b} + \vec{b} \times \vec{c} + \vec{c} \times \vec{a}$ the ordered pair, $(λ, \vec{d})$ is equal to:

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$(\frac{3}{2}, 3 \vec{b} \times \vec{c})$

$(\frac{3}{2}, 3 \vec{a} \times \vec{c})$

$(- \frac{3}{2}, 3 \vec{c} \times \vec{b})$

$(- \frac{3}{2}, 3 \vec{a} \times \vec{b})$

answer is D.

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Detailed Solution

${|\vec{a} + \vec{b} + \vec{c}|}^{2} = 0$

$3 + 2 (\vec{a} . \vec{b} + \vec{b} . \vec{c} + \vec{c} . \vec{a}) = 0$

$(\vec{a} . \vec{b} + \vec{b} . \vec{c} + \vec{c} . \vec{a}) = \frac{- 3}{2}$

$\Rightarrow λ = \frac{- 3}{2}$

$\vec{d} = \vec{a} \times \vec{b} + \vec{b} \times (- \vec{a} - \vec{b}) + (- \vec{a} - \vec{b}) \times \vec{a}$

$= \vec{a} \times \vec{b} + \vec{a} \times \vec{b} + \vec{a} \times \vec{b}$

$\vec{d} = 3 (\vec{a} \times \vec{b})$

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Let a→,b→and c→ be three unit vectors such that a→+b→+c→=0→. If λ=a→.b→+b→.c→+c→.a→and then d→=a→×b→+b→×c→+c→×a→the ordered pair,λ,d→ is equal to:

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