Let a = 2 i ^ - 7 j ^ + 5 k ^ , b = i ^ + k ^ and c = i ^ + 2 j ^ - 3 k ^ be three given vectors. If r is a vector such that r × a = c × a and r · b = 0 , then | r | is equal to [2023]

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Solution

Q.
Let $\vec{a} = 2 \hat{i} - 7 \hat{j} + 5 \hat{k},$ $\vec{b} = \hat{i} + \hat{k}$ and $\vec{c} = \hat{i} + 2 \hat{j} - 3 \hat{k}$ be three given vectors. If $\vec{r}$ is a vector such that $\vec{r} \times \vec{a} = \vec{c} \times \vec{a}$ and $\vec{r} \cdot \vec{b} = 0$ , then $| \vec{r} |$ is equal to [2023]

1 $\frac{11}{7} \sqrt{2}$

2 $\frac{11}{7}$

3 $\frac{\sqrt{914}}{7}$

4 $\frac{11}{5} \sqrt{2}$

Ans.
(1)

$\vec{a} = 2 \hat{i} - 7 \hat{j} + 5 \hat{k},$ $\vec{b} = \hat{i} + \hat{k},$ $\vec{c} = \hat{i} + 2 \hat{j} - 3 \hat{k}$

$\vec{r} \times \vec{a} = \vec{c} \times \vec{a} \Rightarrow (\vec{r} - \vec{c}) \times \vec{a} = 0$

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

$(\hat{i} + 2 \hat{j} - 3 \hat{k}) \cdot (\hat{i} + \hat{k}) + λ (\hat{i} + \hat{k}) \cdot (2 \hat{i} - 7 \hat{j} + 5 \hat{k}) = 0$

$\Rightarrow (1 + 0 - 3) + λ (2 - 0 + 5) = 0;$ $- 2 + λ (7) = 0 \Rightarrow λ = \frac{2}{7}$

$∴$ $\vec{r} = \vec{c} + \frac{2 \vec{a}}{7} = \frac{1}{7} (11 \hat{i} - 11 \hat{k})$ $\Rightarrow$ $| \vec{r} |$ $= \frac{11 \sqrt{2}}{7}$

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