Let , and be three vectors. If a vectors satisfies and , then is equal to [2024]

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Solution

Q.
Let $\vec{a} = 3 \hat{i} + \hat{j} - 2 \hat{k}$ , $\vec{b} = 4 \hat{i} + \hat{j} + 7 \hat{k}$ and $\vec{c} = \hat{i} - 3 \hat{j} + 4 \hat{k}$ be three vectors. If a vectors $\vec{p}$ satisfies $\vec{p} \times \vec{b} = \vec{c} \times \vec{b}$ and $\vec{p} \cdot \vec{a} = 0$ , then $\vec{p} \cdot (\hat{i} - \hat{j} - \hat{k})$ is equal to [2024]

1 28

2 32

3 36

4 24

Ans.
(2)

Given, $\vec{a} = 3 \hat{i} + \hat{j} - 2 \hat{k}$ , $\vec{b} = 4 \hat{i} + \hat{j} + 7 \hat{k}$ and $\vec{c} = \hat{i} - 3 \hat{j} + 4 \hat{k}$

Also, $\vec{p} \times \vec{b} = \vec{c} \times \vec{b}$

$(\vec{p} - \vec{c}) \times \vec{b} = 0 \Rightarrow \vec{b} | | \vec{p} - \vec{c} \Rightarrow (\vec{p} - \vec{c}) = λ \vec{b}$

let $λ \vec{b} + \vec{c} = \vec{p}$

$\Rightarrow \vec{p} = \hat{i} (4 λ + 1) + \hat{j} (λ - 3) + \hat{k} (7 λ + 4) \Rightarrow \vec{p} \cdot \vec{a} = 0$

$\Rightarrow (4 λ + 1) 3 + (λ - 3) (1) + (7 λ + 4) (- 2) = 0$

$\Rightarrow 12 λ + 3 + λ - 3 - 14 λ - 8 \Rightarrow λ = - 8$

$∴ \vec{p} = - 31 \hat{i} - 11 \hat{j} - 52 \hat{k}$

$\vec{p} \cdot (\hat{i} - \hat{j} - \hat{k}) = - 31 + 11 + 52$

$\vec{p} \cdot (\hat{i} - \hat{j} - \hat{k}) = 32$ .

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