The Least positive integral value of , for which the angle between the vectors and is acute, is __________. [2024]

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Solution

Q.
The Least positive integral value of $α$ , for which the angle between the vectors $α \hat{i} - 2 \hat{j} + 2 \hat{k}$ and $α \hat{i} + 2 α \hat{j} - 2 \hat{k}$ is acute, is __________. [2024]

Ans.
(5)

$\cos θ = \frac{\vec{a} \cdot \vec{b}}{| \vec{a} | | \vec{b} |}$

As $θ$ is acute, so $\cos θ > 0$ .

$\frac{(α \hat{i} - 2 \hat{j} + 2 \hat{k}) \cdot (α \hat{i} + 2 α \hat{j} - 2 \hat{k})}{\sqrt{α^{2} + {(- 2)}^{2} + {(2)}^{2}} \times \sqrt{α^{2} + {(2 α)}^{2} + {(- 2)}^{2}}} > 0$

$\Rightarrow \frac{α^{2} - 4 α - 4}{\sqrt{α^{2} + 8} \times \sqrt{5 α^{2} + 4}} > 0$

$\Rightarrow α^{2} - 4 α - 4 > 0 \Rightarrow α^{2} - 4 α + 4 - 4 - 4 > 0$

$\Rightarrow {(α - 2)}^{2} - 8 > 0 \Rightarrow {(α - 2)}^{2} > 8$

$\Rightarrow α - 2 > 2 \sqrt{2} or α - 2 < - 2 \sqrt{2}$ (Neglect)

$\Rightarrow α > 2 + 2 \sqrt{2} = 4.83$

$\Rightarrow$ Least positive integral value of $α = 5$ .

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