Question

If the vectors $\overrightarrow{a}$ and $\overrightarrow{b}$ are such that $\left|\overrightarrow{a}\right|=3,\left|\overrightarrow{b}\right|=\frac{2}{3}$ and $\overrightarrow{a}\times \overrightarrow{b}$ is a unit vector, then write 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 $\overrightarrow{a}\times \overrightarrow{b}$ is a unit vector.

$$\begin{gathered} \therefore \left|\overrightarrow{a}\times \overrightarrow{b}\right|=1 \\ \Rightarrow \left|\overrightarrow{a}\right|\left|\overrightarrow{b}\right|\sin \theta =1 \\ \Rightarrow 3\times \frac{2}{3}\times \sin \theta =1 \\ \Rightarrow \sin \theta =\frac{1}{2} \\ \Rightarrow \theta =\frac{\mathrm{\pi }}{6} \end{gathered}$$

Thus, the angle between $\overrightarrow{a}$ and $\overrightarrow{b}$ is $\frac{\mathrm{\pi }}{6}$.