Question

If $\theta$ be the angle between the vectors $\hat{i}+\hat{j}$ and $\hat{j}+\hat{k}$, then $\theta$ is

• A. $0$
• B. $\pi /4$
• C. $\pi /2$
• D. $\pi /3$

Answer 1

The correct option is D $\pi /3$

$\text{Step 1: Dot product between the vectors}$

Let $\theta$ be the angle between the vectors $\overrightarrow{A}$ and $\overrightarrow{B}$

Dot product between the vectors :

$\overrightarrow{A}.\overrightarrow{B}=|\overrightarrow{A}||\overrightarrow{B}|\cos \theta$

$\Rightarrow \cos \theta =\frac{\overrightarrow{A}.\overrightarrow{B}}{|\overrightarrow{A}||\overrightarrow{B}|}$ $....\left(1\right)$

$\text{Step 2: Angle between the vectors}$

Given : $\overrightarrow{A}=\hat{i}+\hat{j}$ and $\overrightarrow{B}=\hat{j}+\hat{k}$

Putting values of $\overrightarrow{A}$ and $\overrightarrow{B}$ in equation $\left(1\right)$

$\cos \theta =\frac{\overrightarrow{A}.\overrightarrow{B}}{|\overrightarrow{A}||\overrightarrow{B}|}=\frac{(\hat{i}+\hat{j})}{\sqrt{1^2+1^2}}.\frac{(\hat{j}+\hat{k})}{\sqrt{1^2+1^2}}$

$\Rightarrow \cos \theta =\frac{1}{2}$

$\Rightarrow \theta =\frac{\pi }{3}$

Hence the correct option is $D$