Let , . Let a vector be such that the angle between and is and . If , then the value of is equal to [2024]

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Solution

Q.
Let $\vec{a} = \hat{i} + α \hat{j} + β \hat{k}$ , $α, β \in R$ . Let a vector $\vec{b}$ be such that the angle between $\vec{a}$ and $\vec{b}$ is $\frac{π}{4}$ and $| \vec{b} |^{2} = 6$ . If $\vec{a} \cdot \vec{b} = 3 \sqrt{2}$ , then the value of $(α^{2} + β^{2}) | \vec{a} \times \vec{b} |^{2}$ is equal to [2024]

1 95

2 85

3 90

4 75

Ans.
(3)

Given, $\vec{a} \cdot \vec{b} = 3 \sqrt{2}$ and angle between $\vec{a}$ and $\vec{b}$ is $\frac{π}{4}$ .

$\Rightarrow | \vec{a} | | \vec{b} | \cos \frac{π}{4} = 3 \sqrt{2} \Rightarrow | \vec{a} | (\sqrt{6}) (\frac{1}{\sqrt{2}}) = 3 \sqrt{2}$

$\Rightarrow | \vec{a} | = \sqrt{6}$

Since, $\vec{a} = \hat{i} + α \hat{j} + β \hat{k}$ $∴ | \vec{a} | = \sqrt{1 + α^{2} + β^{2}} = \sqrt{6}$

$\Rightarrow 1 + α^{2} + β^{2} = 6 \Rightarrow α^{2} + β^{2} = 5$

also, $| \vec{a} \times \vec{b} |^{2} = | \vec{a} |^{2} | \vec{b} |^{2} \sin^{2} \frac{π}{4} = 6 \times 6 \times \frac{1}{2} = 18$

$∴ (α^{2} + β^{2}) | \vec{a} \times \vec{b} |^{2} = 5 \times 18 = 90$ .

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