If a→×b→ b→×c→ c→×a→=λa→ b→ c→ then λ is equal to__________

a→×b→b→×c→c→×a→=(a→×b→)×(b→×c→)(c→×a→)=(a→×b→.c→)b→−((a→×b→)b→)c→.(c→×a→)=abcb→.(c→×a→)−0=a→b→c→2∴λ=1

If a→×b→ b→×c→ c→×a→=λa→ b→ c→ then λ is equal to__________

a→×b→b→×c→c→×a→=(a→×b→)×(b→×c→)(c→×a→)=(a→×b→.c→)b→−((a→×b→)b→)c→.(c→×a→)=abcb→.(c→×a→)−0=a→b→c→2∴λ=1

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If $[\vec{a} \times \vec{b} \vec{b} \times \vec{c} \vec{c} \times \vec{a}] = λ [\vec{a} \vec{b} \vec{c}]$ then $λ$ is equal to__________

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Detailed Solution

$\begin{array}{l} [\begin{matrix} \vec{a} \times \vec{b} & \vec{b} \times \vec{c} & \vec{c} \times \vec{a} \end{matrix}] \\ = [(\vec{a} \times \vec{b}) \times (\vec{b} \times \vec{c})] (\vec{c} \times \vec{a}) \\ = [(\vec{a} \times \vec{b} . \vec{c}) \vec{b} - ((\vec{a} \times \vec{b}) \vec{b}) \vec{c}] . (\vec{c} \times \vec{a}) \\ = [a b c] \vec{b} . (\vec{c} \times \vec{a}) - 0 \\ = {[\vec{a} \vec{b} \vec{c}]}^{2} \\ ∴ λ = 1 \end{array}$