Question

If $\overrightarrow{a}\mathrm{and}\overrightarrow{b}$are two vectors such that $\left|\overrightarrow{a}\times \overrightarrow{b}\right|=\sqrt{3}\mathrm{and}\overrightarrow{a}.\overrightarrow{b}=1,$ find the angle between.

Answer 1

$$\begin{gathered} \left|\overrightarrow{a}\times \overrightarrow{b}\right|=\sqrt{3} \\ \Rightarrow \left|\overrightarrow{a}\right|\left|\overrightarrow{b}\right|\sin \theta =\sqrt{3}...\left(1\right) \\ \overrightarrow{a}.\overrightarrow{b}=1 \\ \Rightarrow \left|\overrightarrow{a}\right|\left|\overrightarrow{b}\right|\cos \theta =1...\left(2\right) \\ \text{Dividing (1) by (2), we get} \\ \frac{\left|\overrightarrow{a}\right|\left|\overrightarrow{b}\right|\sin \theta }{\left|\overrightarrow{a}\right|\left|\overrightarrow{b}\right|\cos \theta }=\sqrt{3} \\ \Rightarrow \tan \theta =\sqrt{3} \\ \Rightarrow \theta ={60}^o \end{gathered}$$