If c be a non-zero scalar and a→,b→ be given non-zero vectors such thata→⊥b→ find the vector r→ such that a→⋅r→=c, and a→×r→=b→.

r→×a→=−b→ ∴ r→=−a→×b→a→⋅a→+αa→⇒a→⋅r→=c⇒c=αa→⋅a→ ∴ r→=ca→−a→×b→a→⋅a→.

If c be a non-zero scalar and a→,b→ be given non-zero vectors such thata→⊥b→ find the vector r→ such that a→⋅r→=c, and a→×r→=b→.

r→×a→=−b→ ∴ r→=−a→×b→a→⋅a→+αa→⇒a→⋅r→=c⇒c=αa→⋅a→ ∴ r→=ca→−a→×b→a→⋅a→.

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If $c$ be a non-zero scalar and $\vec{a}, \vec{b}$ be given non-zero vectors such that
$\vec{a} ⊥ \vec{b}$ find the vector $\vec{r}$ such that $\vec{a} \cdot \vec{r} = c,$ and $\vec{a} \times \vec{r} = \vec{b}$ .

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$\frac{\vec{b} \cdot \vec{a} - \vec{a} \times \vec{b}}{| \vec{a} |}$

none of these

$\frac{c \vec{a} - \vec{a} \times \vec{b}}{\vec{a} \cdot \vec{a}}$

$\frac{c \vec{a} - \vec{a} \times \vec{b}}{\vec{a}}$

answer is A.

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Detailed Solution

$\vec{r} \times \vec{a} = - \vec{b} ∴ \vec{r} = \frac{- \vec{a} \times \vec{b}}{\vec{a} \cdot \vec{a}} + α \vec{a}$

$\Rightarrow \vec{a} \cdot \vec{r} = c \Rightarrow c = α \vec{a} \cdot \vec{a} ∴ \vec{r} = \frac{c \vec{a} - \vec{a} \times \vec{b}}{\vec{a} \cdot \vec{a}}$ .

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If c be a non-zero scalar and a→,b→ be given non-zero vectors such thata→⊥b→ find the vector r→ such that a→⋅r→=c, and a→×r→=b→.

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If $c$ be a non-zero scalar and $\vec{a}, \vec{b}$ be given non-zero vectors such that
$\vec{a} ⊥ \vec{b}$ find the vector $\vec{r}$ such that $\vec{a} \cdot \vec{r} = c,$ and $\vec{a} \times \vec{r} = \vec{b}$ .