If a→ and b→ are orthogonal unit vectors, then for a vector r→ non-coplanar with a→ and b→ vector r→×a→ is equal to

r→×a→=λa→+μb→+γa→×b→∴ [r→a→a→]=λa→⋅a→+μb→⋅a→+γ[a→b→a→] 0=λ|a→|2+0+0 λ=0also [r→a→b→]=λa→⋅b→+μb→⋅b→+γ[a→b→b→]=μalso (r→×a→)×b→=γ(a→×b→)×b→or (r→⋅b→)a→−(a→⋅b→)r→=γ{(a→⋅b→)b→−(b→⋅b→)a→}or (r→⋅b→)a→=−γa→ or γ=−(r→⋅b→)

If a→ and b→ are orthogonal unit vectors, then for a vector r→ non-coplanar with a→ and b→ vector r→×a→ is equal to

r→×a→=λa→+μb→+γa→×b→∴ [r→a→a→]=λa→⋅a→+μb→⋅a→+γ[a→b→a→] 0=λ|a→|2+0+0 λ=0also [r→a→b→]=λa→⋅b→+μb→⋅b→+γ[a→b→b→]=μalso (r→×a→)×b→=γ(a→×b→)×b→or (r→⋅b→)a→−(a→⋅b→)r→=γ{(a→⋅b→)b→−(b→⋅b→)a→}or (r→⋅b→)a→=−γa→ or γ=−(r→⋅b→)

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If $\vec{a} and \vec{b}$ are orthogonal unit vectors, then for a vector $\vec{r}$ non-coplanar with $\vec{a} and \vec{b}$ vector $\vec{r} \times \vec{a}$ is equal to

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$[\vec{r} \vec{a} \vec{b}] \vec{b} - (\vec{r} \cdot \vec{b}) (\vec{b} \times \vec{a})$

none of these

$[\vec{r} \vec{a} \vec{b}] (\vec{a} + \vec{b})$

$[\vec{r} \vec{a} \vec{b}] \vec{a} + (\vec{r} \cdot \vec{a}) \vec{a} \times \vec{b}$

answer is A.

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Detailed Solution

$\begin{array}{l} \vec{r} \times \vec{a} = λ \vec{a} + μ \vec{b} + γ \vec{a} \times \vec{b} \\ ∴ [\vec{r} \vec{a} \vec{a}] = λ \vec{a} \cdot \vec{a} + μ \vec{b} \cdot \vec{a} + γ [\vec{a} \vec{b} \vec{a}] \\ 0 = λ | \vec{a} |^{2} + 0 + 0 \\ λ = 0 \end{array}$

$\begin{array}{l} also [\vec{r} \vec{a} \vec{b}] = λ \vec{a} \cdot \vec{b} + μ \vec{b} \cdot \vec{b} + γ [\vec{a} \vec{b} \vec{b}] = μ \\ also (\vec{r} \times \vec{a}) \times \vec{b} = γ (\vec{a} \times \vec{b}) \times \vec{b} \end{array}$

$\begin{matrix} or (\vec{r} \cdot \vec{b}) \vec{a} - (\vec{a} \cdot \vec{b}) \vec{r} = γ {(\vec{a} \cdot \vec{b}) \vec{b} - (\vec{b} \cdot \vec{b}) \vec{a}} \\ or (\vec{r} \cdot \vec{b}) \vec{a} = - γ \vec{a} or γ = - (\vec{r} \cdot \vec{b}) \end{matrix}$