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

<p>Given $\vec{a}=\alpha t \hat{i}+6 \hat{j}-3 \hat{k}$</p> <p>and $\vec{b}=t \hat{i}-2 \hat{j}-2 \alpha t \hat{k}$</p> <p>angle between $\vec{a}$ and $\vec{b}$ is given by</p> <p>$\cos \theta=\frac{\vec{a} \cdot \vec{b}}{|\vec{a}||\vec{b}|}$</p> <p>We have, $\cos \theta < 0(\because$ angle between $\vec{a}$ and $\vec{b}$ is obtuse)</p> <p>$$\begin{aligned} & \Rightarrow \quad \vec{a} \cdot \vec{b}<0 \\ & \Rightarrow \alpha t^2-12+6 \alpha t<0 \forall t \in \mathbb{R} \end{aligned}$$</p> <p>If $\alpha=0$, then $-12<0$ (condition holds)</p> <p>If $\alpha \neq 0 \Rightarrow \alpha<0\quad \text{.... (i)}$</p> <p>And maximum value of $\alpha t^2+6 \alpha t-12<0$</p> <p>$$\begin{aligned} & \Rightarrow \frac{-D}{4 a}<0 \text { (where } D \text { is discriminant and } a=\alpha) \\ & \Rightarrow \frac{36 \alpha^2+48 \alpha}{4 \alpha}>0 \\ & \Rightarrow \alpha>\frac{-4}{3} \\ & \therefore \alpha \in\left(\frac{-4}{3}, 0\right] \end{aligned}$$</p>