Question

Find the values of $\lambda$ for which the angle between the vectors the vectors $\overrightarrow{a}=2\lambda \widehat{{}^{2}i+}4\lambda \hat{j}+\hat{k}$ and $\overrightarrow{b}=\widehat{7i-}2\hat{j}+\widehat{\lambda k}$ is obtuse.

Answer 1

$\cos \theta =\frac{\overrightarrow{a}.\overrightarrow{b}}{|\overrightarrow{a}.\overrightarrow{b}|}$

$=\frac{(2{\lambda }^2\hat{i}+4\lambda \hat{j}+\hat{k}).(7\hat{i}-2\hat{j}+\lambda \hat{k})}{\sqrt{4{\lambda }^4+16{\lambda }^2+1}\sqrt{49+4+{\lambda }^2}}$

$=\frac{14{\lambda }^2-8\lambda +\lambda }{\sqrt{4{\lambda }^4+16{\lambda }^2}+1\sqrt{49+4+{\lambda }^2}}$

$\frac{7(2{\lambda }^2-\lambda )}{\sqrt{}+1\sqrt{49+4+{\lambda }^2}}$

$-1\le \cos \theta \le 0$ for $\theta$ to be $\theta b$

$\cos \theta \le 0$

$\Rightarrow \lambda (2\lambda -1)\le 0$

$\Rightarrow \lambda \in [0,1/2]$ AS $|\cos \theta |<1$