The set of all , for which the vectors a= ti+6j-3k and b=ti-2j-2 tk are inclined at an obt…

Question

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 $t\in R$, is

Accepted Answer

Given $\vec{a}=\alpha t\hat{i}+6\hat{j}-3\hat{k}$

and $\vec{b}=t\hat{i}-2\hat{j}-2\alpha t\hat{k}$

angle between $\vec{a}$ and $\vec{b}$ is given by

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

We have, $\cos 	heta $ \& \Rightarrow \, \vec{a}\cdot \vec{b}If $\alpha =0$, then $-12If $\alpha 
e 0\Rightarrow \alpha And maximum value of $\alpha {t}^{2}+6\alpha t-12$

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