Question

If $\overrightarrow{a}\mathrm{and}\overrightarrow{b}$are unit vectors such that $\overrightarrow{a}\times \overrightarrow{b}$is also a unit vector, find the angle between $\overrightarrow{a}\mathrm{and}\overrightarrow{b}$.

Answer 1

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