Question

The values of $x$ for which the angle between $\overrightarrow{a}=2x^2\hat{i}+4x\hat{j}+\hat{k}$and $\overrightarrow{b}=7\hat{i}-2\hat{j}+x\hat{k}$ is obtuse?

• A. $0<x<\frac{1}{2}$
• B. $\frac{1}{2}<x<15$
• C. $x>\frac{1}{2}$ or $x<0$
• D. Form a null set

Answer 1

The correct option is A

$0<x<\frac{1}{2}$

Explanation for correct option:

Given, $\overrightarrow{a}=2x^2\hat{i}+4x\hat{j}+\hat{k}$and $\overrightarrow{b}=7\hat{i}-2\hat{j}+x\hat{k}$

Angle between two vector $\overrightarrow{a}$ and $\overrightarrow{b}$ is given by

$\cos \left(\theta \right)=\frac{\overrightarrow{a}·\overrightarrow{b}}{\left|\overrightarrow{a}\right|·\left|\overrightarrow{b}\right|}$

Now, if the angle is obtuse, then $\cos \left(\theta \right)<0$

$$\begin{gathered} \therefore \frac{\overrightarrow{a}·\overrightarrow{b}}{\left|\overrightarrow{a}\right|·\left|\overrightarrow{b}\right|}<0 \\ \Rightarrow \overrightarrow{a}·\overrightarrow{b}<0 \\ \Rightarrow 14x^2-8x+x<0 \\ \Rightarrow 7x\left(2x-1\right)<0 \\ \therefore 0<x<\frac{1}{2} \end{gathered}$$

Hence option A is correct.