Question

If $\overrightarrow{a}\mathrm{and}\overrightarrow{b}$are two vectors of magnitudes 3 and $\frac{\sqrt{2}}{3}$ respectively such that $\overrightarrow{a}\times \overrightarrow{b}$is a unit vector. Write the angle between $\overrightarrow{a}\mathrm{and}\overrightarrow{b}.$

Answer 1

$$\begin{gathered} \text{Let}\mathit{\text{θ}}\text{be the angle between}\overrightarrow{a}\text{and}\overrightarrow{b}\text{.} \\ \text{It is given that}\overrightarrow{a}\times \overrightarrow{b}\text{is a unit vector.} \\ \Rightarrow \left|\overrightarrow{a}\times \overrightarrow{b}\right|=1 \\ \text{We know} \\ \left|\overrightarrow{a}\times \overrightarrow{b}\right|=\left|\overrightarrow{a}\right|\left|\overrightarrow{b}\right|\sin \theta \\ \Rightarrow 1=\left(3\right)\left(\frac{\sqrt{2}}{3}\right)\sin \theta \\ \Rightarrow \sin \theta =\frac{1}{\sqrt{2}} \\ \Rightarrow \theta ={45}^o,{135}^o \end{gathered}$$