The set of all $\alpha$, for which the vectors $\vec{a}=\alpha t \hat{i}+6 \hat{j}-3 \hat{k}$ and $\vec{b}=t \hat{i}-2 \hat{j}-2 \alpha t \hat{k}$ are inclined at an obtuse angle for all $t \in \mathbb{R}$, is

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Aicharya · Accepted Answer

$\begin{aligned} & \overrightarrow{\mathrm{a}}=\alpha \hat{\mathrm{i}}+6 \hat{\mathrm{j}}-3 \hat{\mathrm{k}} \ & \overrightarrow{\mathrm{b}}=\mathrm{t} \hat{\mathrm{i}}-2 \hat{\mathrm{j}}-2 \alpha \hat{\mathrm{k}} \ & 	ext { so } \overrightarrow{\mathrm{a}} \overrightarrow{\mathrm{b}} < 0, \forall \mathrm{t} \in \mathrm{R} \ & \alpha \mathrm{t}^2-12+6 \alpha \mathrm{t} < 0 \ & \alpha \mathrm{t}^2+6 \alpha \mathrm{t}-12 < 0, \forall \mathrm{t} \in \mathrm{R} \ & \alpha < 0 	ext {, and } \mathrm{D} < 0 \ & 36 \alpha^2+48 \alpha < 0 \ & 12 \alpha(3 \alpha+4) < 0 \ & \frac{-4}{3} < \alpha < 0\end{aligned}$ also for $\mathrm{a}=0, \overrightarrow{\mathrm{a}} \cdot \overrightarrow{\mathrm{b}} < 0$ hence a $\alpha \in\left(\frac{-4}{3}, 0ight]$

The set of all $\alpha$, for which the vectors $\vec{a}=\alpha t \hat{i}+6 \hat{j}-3 \hat{k}$ and $\vec{b}=t…

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