Let , , and be a vector such that and . Then is equal to __________. [2025]

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Solution

Q.
Let $\vec{a} = \hat{i} + 2 \hat{j} + \hat{k}$ , $\vec{b} = 3 \hat{i} - 3 \hat{j} + 3 \hat{k}$ , $\vec{c} = 2 \hat{i} - \hat{j} + 2 \hat{k}$ and $\vec{d}$ be a vector such that $\vec{b} \times \vec{d} = \vec{c} \times \vec{d}$ and $\vec{a} \cdot \vec{d} = 4$ . Then $| (\vec{a} \times \vec{d}) |^{2}$ is equal to __________. [2025]

Ans.
(128)

Given, $\vec{b} \times \vec{d} = \vec{c} \times \vec{d}$ and $\vec{a} \cdot \vec{d} = 4$

$\Rightarrow \vec{d} = λ (\vec{b} - \vec{c}) = λ (\hat{i} - 2 \hat{j} + \hat{k})$ $∵ \vec{a} \cdot \vec{d} = 4 \Rightarrow λ = - 2$

Also, $| \vec{a} \times \vec{d} |^{2} + | \vec{a} \cdot \vec{d} |^{2} = | \vec{a} |^{2} | \vec{d} |^{2}$

$\Rightarrow | \vec{a} \times \vec{d} |^{2} = 6 \times 24 - 16 = 128$ .

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