Question

Let $\overrightarrow{a}=x^2\hat{i}+2\hat{j}-2\hat{k},\overrightarrow{b}=\hat{i}-\hat{j}+\hat{k}\mathrm{and}\overrightarrow{c}=x^2\hat{i}+5\hat{j}-4\hat{k}$ be three vectors. Find the values of x for which the angle between $\overrightarrow{a}\mathrm{and}\overrightarrow{b}$is acute and the angle between $\overrightarrow{b}\mathrm{and}\overrightarrow{c}$is obtuse.

Answer 1

$$\begin{gathered} \mathrm{We}\mathrm{have} \\ \overrightarrow{a}==x^2\stackrel{⏜}{i}+2\hat{j}-2\hat{k},\overrightarrow{b}=\hat{i}-\hat{j}+\hat{k}\mathrm{and}\overrightarrow{c}=x^2\hat{i}+5\hat{j}-4\hat{k} \\ {\text{Let θ}}_1\text{be the angle between}\overrightarrow{a}\text{and}\overrightarrow{b}{\text{and θ}}_2\text{be the angle between}\overrightarrow{b}\text{and}\overrightarrow{c}\text{.} \\ {\text{Given that θ}}_1{\text{is acute and θ}}_2\text{is obtuse.} \\ \Rightarrow \cos {\theta }_1>0\mathrm{and}\cos {\theta }_2<0 \\ \Rightarrow \frac{\overrightarrow{a}.\overrightarrow{b}}{\left|\overrightarrow{a}\right|.\left|\overrightarrow{b}\right|}>o\mathrm{and}\frac{\overrightarrow{b}.\overrightarrow{c}}{\left|\overrightarrow{b}\right|.\overrightarrow{\left|c\right|}}<0 \\ \Rightarrow \frac{x^2-4}{\sqrt{x^4+4+4}\sqrt{1+1+1}}>0\mathrm{and}\frac{x^2-9}{\sqrt{1+1+1}\sqrt{x^4+25+16}}<0 \\ \Rightarrow x^2-4>0\mathrm{and}{x}^2-9<0 \\ \Rightarrow x\in \left(-\infty ,-2\right)\cup \left(2,\infty \right)\mathrm{and}x\in \left(-3,3\right) \\ \Rightarrow x\in \left(-3,-2\right)\cup \left(2,3\right) \end{gathered}$$