Let and . Let be a unit vector in the plane of the vectors and and be perpendicular to . Then such a vector is : [2025]

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Solution

Q.
Let $\vec{a} = \hat{i} + 2 \hat{j} + \hat{k}$ and $\vec{b} = 2 \hat{i} + \hat{j} - \hat{k}$ . Let $\hat{c}$ be a unit vector in the plane of the vectors $\vec{a}$ and $\vec{b}$ and be perpendicular to $\vec{a}$ . Then such a vector $\hat{c}$ is : [2025]

1 $\frac{1}{\sqrt{2}} (- \hat{i} + \hat{k})$

2 $\frac{1}{\sqrt{3}} (\hat{i} - \hat{j} + \hat{k})$

3 $\frac{1}{\sqrt{3}} (- \hat{i} + \hat{j} - \hat{k})$

4 $\frac{1}{\sqrt{5}} (\hat{j} - 2 \hat{k})$

Ans.
(1)

We have, $\vec{a} = \hat{i} + 2 \hat{j} + \hat{k}$ , $\vec{b} = 2 \hat{i} + \hat{j} - \hat{k}$

Let $\vec{c} = λ \vec{a} + μ \vec{b}$

Now, $\vec{c} \cdot \vec{a} = 0$

$\Rightarrow (λ \vec{a} + μ \vec{b}) \cdot \vec{a} = 0 \Rightarrow λ \vec{a} \cdot \vec{a} + μ \vec{b} \cdot \vec{a} = 0$

$\Rightarrow λ (6) + μ (3) = 0 \Rightarrow μ = - 2 λ$

$∴ \vec{c} = λ \vec{a} - 2 λ \vec{b}$

$= λ (\hat{i} + 2 \hat{j} + \hat{k} - 2 (2 \hat{i} + \hat{j} - \hat{k}))$

$= λ (- 3 \hat{i} + 3 \hat{k})$

Since, $\vec{c}$ is a unit vector.

$\Rightarrow$ $| λ (- 3 \hat{i} + 3 \hat{k}) |$ $= 1 \Rightarrow$ $| λ |$ $\sqrt{9 + 9} = 1$

$\Rightarrow λ = \pm \frac{1}{3 \sqrt{2}} \Rightarrow \hat{c} = \pm \frac{1}{\sqrt{2}} (- \hat{i} + \hat{k})$ .

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