Question

If $\overrightarrow{a}$ and $\overrightarrow{b}$ are two unit vectors such that $\overrightarrow{a}+\overrightarrow{b}$ is also a unit vector, then find the angle between $\overrightarrow{a}$ and $\overrightarrow{b}$. [CBSE 2014]

Answer 1

Let the angle between $\overrightarrow{a}$ and $\overrightarrow{b}$ be $\theta$.

It is given that $\left|\overrightarrow{a}\right|=\left|\overrightarrow{b}\right|=\left|\overrightarrow{a}+\overrightarrow{b}\right|=1$.

$$\begin{gathered} \left|\overrightarrow{a}+\overrightarrow{b}\right|=1 \\ \Rightarrow {\left|\overrightarrow{a}+\overrightarrow{b}\right|}^2=1 \\ \Rightarrow {\left|\overrightarrow{a}\right|}^2+2\left|\overrightarrow{a}\right|\left|\overrightarrow{b}\right|\cos \theta +{\left|\overrightarrow{b}\right|}^2=1 \\ \Rightarrow 1+2\times 1\times 1\times \cos \theta +1=1 \\ \Rightarrow 2\cos \theta =-1 \\ \Rightarrow \cos \theta =-\frac{1}{2}=\cos \frac{2\mathrm{\pi }}{3} \\ \Rightarrow \theta =\frac{2\mathrm{\pi }}{3} \end{gathered}$$

Thus, the angle between $\overrightarrow{a}$ and $\overrightarrow{b}$ is $\frac{2\mathrm{\pi }}{3}$.