Question

If $\overrightarrow{a}\mathrm{and}\overrightarrow{b}$are unit vectors inclined at an angle θ, then the value of $\left|\overrightarrow{a}-\overrightarrow{b}\right|$ is
(a) $2\sin \frac{\mathrm{\theta }}{2}$

(b) 2 sin θ

(c) $2\cos \frac{\mathrm{\theta }}{2}$

(d) 2 cos θ

Answer 1

(a) $2\sin \frac{\mathrm{\theta }}{2}$

$$\begin{gathered} \overrightarrow{a}.\overrightarrow{b}=\left|\overrightarrow{a}\right|\left|\overrightarrow{b}\right|\cos \theta \\ =1\times 1\cos \theta \text{(Because}\overrightarrow{a}\text{and}\overrightarrow{b}\text{are unit vectors)} \\ =\cos \theta ...\left(\mathrm{i}\right) \\ {\left|\overrightarrow{a}-\overrightarrow{b}\right|}^2={\left|\overrightarrow{a}\right|}^2+{\left|\overrightarrow{b}\right|}^2-2\overrightarrow{a}.\overrightarrow{b} \\ =1+1-2\cos \theta \left[\mathrm{Using}\left(\mathrm{i}\right)\right] \\ =2-2\cos \theta \\ =2\left(1-\cos \theta \right) \\ =2\left(2{\sin }^2\frac{\theta }{2}\right) \\ =4{\sin }^2\frac{\theta }{2} \\ \therefore \left|\overrightarrow{a}-\overrightarrow{b}\right|=2\sin \frac{\theta }{2} \end{gathered}$$