Let , and be a vector such that and . Then the maximum value of is : [2025]

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Solution

Q.
Let $\vec{a} = 2 \hat{i} - \hat{j} + 3 \hat{k}$ , $\vec{b} = 3 \hat{i} - 5 \hat{j} + \hat{k}$ and $\vec{c}$ be a vector such that $\vec{a} \times \vec{c} = \vec{c} \times \vec{b}$ and $(\vec{a} + \vec{c}) \cdot (\vec{b} + \vec{c}) = 168$ . Then the maximum value of $| \vec{c} |^{2}$ is : [2025]

1 462

2 308

3 77

4 154

Ans.
(2)

We have, $\vec{a} \times \vec{c} = \vec{c} \times \vec{b}$

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

$\Rightarrow \vec{c} = λ (\vec{a} + \vec{b}) = λ (5 \hat{i} - 6 \hat{j} + 4 \hat{k})$

Also, $(\vec{a} + \vec{c}) \cdot (\vec{b} + \vec{c}) = 168$

$\Rightarrow \vec{a} \cdot \vec{b} + \vec{c} \cdot \vec{b} + \vec{a} \cdot \vec{c} + \vec{c} \cdot \vec{c} = 168$

$\Rightarrow 14 + \vec{c} \cdot (\vec{a} + \vec{b}) + | \vec{c} |^{2} = 168$

$\Rightarrow 14 + λ \cdot 77 + λ^{2} \cdot 77 = 168$

$\Rightarrow 77 λ^{2} + 77 λ - 154 = 0$

$\Rightarrow λ^{2} + λ - 2 = 0 \Rightarrow λ = - 2, 1$

Maximum value of $| \vec{c} |^{2}$ occurs when $λ = - 2$

$| \vec{c} |^{2} = 77 λ^{2} = 77 \times 4 = 308$ .

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