Let a ? = 9 i ^ – 13 j ^ + 25 k ^ , b ? = 3 i ^ + 7 j ^ – 13 k ^ and c ? = 17 i ^ – 2 j ^ + k ^ be three given vectors. If r ? is a vector such that r ? × a ? = ( b ? + c ? ) × a ? and r ? · ( b ? – c ? ) = 0 , then | 593 r ? + 67 a ? | 2 ( 593

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Solution

Q.
Let $\vec{a} = 9 \hat{i} - 13 \hat{j} + 25 \hat{k}$ , $\vec{b} = 3 \hat{i} + 7 \hat{j} - 13 \hat{k}$ and $\vec{c} = 17 \hat{i} - 2 \hat{j} + \hat{k}$ be three given vectors. If $\vec{r}$ is a vector such that $\vec{r} \times \vec{a} = (\vec{b} + \vec{c}) \times \vec{a}$ and $\vec{r} \cdot (\vec{b} - \vec{c}) = 0$ , then $\frac{| 593 \vec{r} + 67 \vec{a} |^{2}}{{(593)}^{2}}$ is equal to __________. [2024]

Ans.
(569)

We have, $\vec{r} \times \vec{a} = (\vec{b} + \vec{c}) \times \vec{a}$

$\Rightarrow [\vec{r} - (\vec{b} + \vec{c})] \times \vec{a} = 0 \Rightarrow \vec{r} - (\vec{b} + \vec{c}) = λ \vec{a}$ , for some scalar $λ$ .

$\Rightarrow \vec{r} = λ \vec{a} + \vec{b} + \vec{c}$

Also, $\vec{r} \cdot (\vec{b} - \vec{c}) = 0 \Rightarrow (λ \vec{a} + \vec{b} + \vec{c}) \cdot (\vec{b} - \vec{c}) = 0$

$\Rightarrow λ \vec{a} \cdot \vec{b} - λ \vec{a} \cdot \vec{c} + | \vec{b} |^{2} - \vec{b} \cdot \vec{c} + \vec{b} \cdot \vec{c} - | \vec{c} |^{2} = 0$

$\Rightarrow λ = \frac{| \vec{c} |^{2} - | \vec{b} |^{2}}{\vec{a} \cdot \vec{b} - \vec{a} \cdot \vec{c}} = \frac{294 - 227}{- 389 - 204} = \frac{- 67}{593}$

$\Rightarrow \vec{r} = \vec{b} + \vec{c} - \frac{67}{593} \vec{a} ∴ \frac{| 593 \vec{r} + 67 \vec{a} |^{2}}{{(593)}^{2}} = \frac{(593 {(\vec{b} + \vec{c}))}^{2}}{{(593)}^{2}}$

$= | \vec{b} + \vec{c} |^{2} = | 20 \hat{i} + 5 \hat{j} - 12 \hat{k} |^{2} = 569$ .

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