Let a ? = 4 i ^ – j ^ + k ^ , b ? = 11 i ^ – j ^ + k ^ and c ? be a vector that ( a ? + b ? ) × c ? = c ? × ( – 2 a ? + 3 b ? ) . If ( 2 a ? + 3 b ? ) · c ? = 1670 , then | c ? | 2 is equal to : [2024]

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Solution

Q.
Let $\vec{a} = 4 \hat{i} - \hat{j} + \hat{k}$ , $\vec{b} = 11 \hat{i} - \hat{j} + \hat{k}$ and $\vec{c}$ be a vector that $(\vec{a} + \vec{b}) \times \vec{c} = \vec{c} \times (- 2 \vec{a} + 3 \vec{b})$ . If $(2 \vec{a} + 3 \vec{b}) \cdot \vec{c} = 1670$ , then $| \vec{c} |^{2}$ is equal to : [2024]

1 1600

2 1618

3 1627

4 1609

Ans.
(2)

We have, $\vec{a} = 4 \hat{i} - \hat{j} + \hat{k}$ and $\vec{b} = 11 \hat{i} - \hat{j} + \hat{k}$

$\vec{a} \cdot \vec{b} = 44 + 1 + 1 = 46, | \vec{a} |^{2} = 18, | \vec{b} |^{2} = 123$

$(\vec{a} + \vec{b}) \times \vec{c} = \vec{c} \times (- 2 \vec{a} + 3 \vec{b})$ ... (i) [Given]

and $(2 \vec{a} + 3 \vec{b}) \cdot \vec{c} = 1670$ ... (ii)

$∴ (\vec{a} + \vec{b}) \times \vec{c} = (2 \vec{a} - 3 \vec{b}) \times \vec{c}$ [From (i)]

$\Rightarrow (- \vec{a} + 4 \vec{b}) \times \vec{c} = 0 \Rightarrow \vec{c} = λ (4 \vec{b} - \vec{a})$

$\Rightarrow (2 \vec{a} + 3 \vec{b}) \cdot λ (4 \vec{b} - \vec{a}) = 1670$ [From (ii)]

$\Rightarrow λ (5 \vec{a} \cdot \vec{b} - 2 | \vec{a} |^{2} + 12 | \vec{b} |^{2}) = 1670$

$\Rightarrow λ = \frac{1670}{5 \times 46 - 2 \times 18 + 12 \times 123} = 1$

Now, $\vec{c} = 4 \vec{b} - \vec{a} = 4 (11 \hat{i} - \hat{j} + \hat{k}) - (4 \hat{i} - \hat{j} + \hat{k})$

$= 40 \hat{i} - 3 \hat{j} + 3 \hat{k}$

$∴ | \vec{c} |^{2} = 1600 + 9 + 9 = 1618$ .

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