Let a ? = 2 i ^ + 5 j ^ – k ^ , b ? = 2 i ^ – 2 j ^ + 2 k ^ and c ? be three vectors such that ( c ? + i ^ ) × ( a ? + b ? + i ^ ) = a ? × ( c ? + i ^ ) . If a ? · c ? = – 29 , then c ? · ( – 2 i ^ + j ^ + k ^ ) is equal to : [2024]

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Solution

Q.
Let $\vec{a} = 2 \hat{i} + 5 \hat{j} - \hat{k}$ , $\vec{b} = 2 \hat{i} - 2 \hat{j} + 2 \hat{k}$ and $\vec{c}$ be three vectors such that $(\vec{c} + \hat{i}) \times (\vec{a} + \vec{b} + \hat{i}) = \vec{a} \times (\vec{c} + \hat{i})$ . If $\vec{a} \cdot \vec{c} = - 29$ , then $\vec{c} \cdot (- 2 \hat{i} + \hat{j} + \hat{k})$ is equal to : [2024]

1 15

2 5

3 12

4 10

Ans.
(2)

Consider $\vec{a} + \vec{b} + \hat{i}$

$= 2 \hat{i} + 5 \hat{j} - \hat{k} + 2 \hat{i} - 2 \hat{j} + 2 \hat{k} + \hat{i}$

$= 5 \hat{i} + 3 \hat{j} + \hat{k}$

Now, $(\vec{c} + \hat{i}) \times (\vec{a} + \vec{b} + \hat{i}) = \vec{a} \times (\vec{c} + \hat{i})$

$\Rightarrow (\vec{c} + \hat{i}) \times (5 \hat{i} + 3 \hat{j} + \hat{k}) = (2 \hat{i} + 5 \hat{j} - \hat{k}) \times (\vec{c} + \hat{i})$

$\Rightarrow (\vec{c} + \hat{i}) \times (5 \hat{i} + 3 \hat{j} + \hat{k} + 2 \hat{i} + 5 \hat{j} - \hat{k}) = 0$

$\Rightarrow (\vec{c} + \hat{i}) \times (7 \hat{i} + 8 \hat{j}) = 0$

$\Rightarrow \vec{c} + \hat{i} = λ (7 \hat{i} + 8 \hat{j})$

$\vec{a} \cdot \vec{c} + \vec{a} \cdot \hat{i} = λ \vec{a} \cdot (7 \hat{i} + 8 \hat{j})$

$\Rightarrow - 29 + 2 = λ (14 + 40) \Rightarrow λ = \frac{- 1}{2}$

$∴ \vec{c} + \hat{i} = \frac{- 7}{2} \hat{i} - 4 \hat{j}$

$\Rightarrow \vec{c} = \frac{- 9}{2} \hat{i} - 4 \hat{j}$

$∴ \vec{c} \cdot (- 2 \hat{i} + \hat{j} + \hat{k}) = 9 - 4 = 5$ .

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