Question

If $\overrightarrow{a}\mathrm{and}\overrightarrow{b}$be two unit vectors and θ the angle between them, then $\overrightarrow{a}+\overrightarrow{b}$is a unit vector if θ =
(a) $\frac{\mathrm{\pi }}{4}$

(b) $\frac{\mathrm{\pi }}{3}$

(c) $\frac{\mathrm{\pi }}{2}$

(d) $\frac{2\mathrm{\pi }}{3}$

Answer 1

(d) $\frac{2\mathrm{\pi }}{3}$

$$\begin{gathered} \mathrm{We}\mathrm{have} \\ \left|\overrightarrow{a}\right|=1\mathrm{and}\left|\overrightarrow{b}\right|=1 \\ \mathrm{Now},\left|\overrightarrow{a}+\overrightarrow{b}\right|=1 \\ \Rightarrow {\left|\overrightarrow{a}\right|}^2+{\left|\overrightarrow{b}\right|}^2+2\overrightarrow{a}.\overrightarrow{b}=1 \\ \Rightarrow 1+1+2\left|\overrightarrow{a}\right|\left|\overrightarrow{b}\right|\cos \theta =1 \\ \Rightarrow 2+2\cos \theta =1 \\ \Rightarrow 2\cos \theta =-1 \\ \Rightarrow \cos \theta =\frac{-1}{2} \\ \Rightarrow \theta =\frac{2\mathrm{\pi }}{3} \end{gathered}$$