Question

Vectors $\overrightarrow{a}\mathrm{and}\overrightarrow{b}$are such that $\left|\overrightarrow{a}\right|=\sqrt{3},\left|\overrightarrow{b}\right|=\frac{2}{3}\mathrm{and}\left(\overrightarrow{a}\times \overrightarrow{b}\right)$ is a unit vector. Write the angle between $\overrightarrow{a}\mathrm{and}\overrightarrow{b}$.

Answer 1

$$\begin{gathered} \text{Given:}\overrightarrow{a}\times \overrightarrow{b}\text{is a unit vector.} \\ \Rightarrow \left|\overrightarrow{a}\times \overrightarrow{b}\right|=1...\left(1\right) \\ \text{Let}\mathit{\text{θ}}\text{be the angle between}\overrightarrow{a}\text{and}\overrightarrow{b}. \\ \text{We know} \\ \left|\overrightarrow{a}\times \overrightarrow{b}\right|=\left|\overrightarrow{a}\right|\left|\overrightarrow{b}\right|\sin \theta \\ \text{From (1), we get} \\ 1=\left(\sqrt{3}\right)\left(\frac{2}{3}\right)\sin \theta \\ \Rightarrow \sin \theta =\frac{\sqrt{3}}{2} \\ \Rightarrow \theta =\frac{\mathrm{\pi }}{3} \end{gathered}$$