Let a = i ^ + 2 j ^ + 3 k ^ , b = i ^ - j ^ + 2 k ^ and c = 5 i ^ - 3 j ^ + 3 k ^ be three vectors. If r is a vector such that r × b ? = c × b ? and r · a = 0 , then 25 | r | 2 is equal to [2023]

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Solution

Q.
Let $\vec{a} = \hat{i} + 2 \hat{j} + 3 \hat{k},$ $\vec{b} = \hat{i} - \hat{j} + 2 \hat{k}$ and $\vec{c} = 5 \hat{i} - 3 \hat{j} + 3 \hat{k}$ be three vectors. If $\vec{r}$ is a vector such that $\vec{r} \times \vec{b} = \vec{c} \times \vec{b}$ and $\vec{r} \cdot \vec{a} = 0$ , then $25$ $| \vec{r} |^{2}$ is equal to [2023]

1 560

2 339

3 449

4 336

Ans.
(2)

Given, $\vec{a} = \hat{i} + 2 \hat{j} + 3 \hat{k},$ $\vec{b} = \hat{i} - \hat{j} + 2 \hat{k}$ and $\vec{c} = 5 \hat{i} - 3 \hat{j} + 3 \hat{k}$

$\vec{r} \times \vec{b} = \vec{c} \times \vec{b}$ and $\vec{r} \cdot \vec{a} = 0 \Rightarrow \vec{r} - \vec{c} = λ \vec{b} \Rightarrow \vec{r} = \vec{c} + λ \vec{b}$

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

$∴$ $λ = \frac{- \vec{a} \cdot \vec{c}}{\vec{a} \cdot \vec{b}} = \frac{- (\hat{i} + 2 \hat{j} + 3 \hat{k}) \cdot (5 \hat{i} - 3 \hat{j} + 3 \hat{k})}{(\hat{i} + 2 \hat{j} + 3 \hat{k}) \cdot (\hat{i} - \hat{j} + 2 \hat{k})} = \frac{- 8}{5}$

$\vec{r} = 5 \hat{i} - 3 \hat{j} + 3 \hat{k} - \frac{8}{5} (\hat{i} - \hat{j} + 2 \hat{k})$

$\Rightarrow \vec{r} = \frac{5 (5 \hat{i} - 3 \hat{j} + 3 \hat{k}) - 8 (\hat{i} - \hat{j} + 2 \hat{k})}{5}$

$\Rightarrow \vec{r} = \frac{17 \hat{i} - 7 \hat{j} - 1 \hat{k}}{5}$ $∴$ $| \vec{r} |^{2}$ $= \frac{1}{25} (289 + 50)$

$\Rightarrow 25$ $| \vec{r} |^{2}$ $= 339$

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