Question

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

Answer 1

$$\begin{gathered} \text{Let}\mathit{\text{θ}}\text{be the angle between}\overrightarrow{a}\text{and}\overrightarrow{b}. \\ \text{We know} \\ \left|\overrightarrow{a}\times \overrightarrow{b}\right|=\left|\overrightarrow{a}\right|\left|\overrightarrow{b}\right|\left|\sin \theta \right| \\ \left|\overrightarrow{a}.\overrightarrow{b}\right|=\left|\overrightarrow{a}\right|\left|\overrightarrow{b}\right|\left|\cos \theta \right| \\ \mathrm{Now}, \\ \left|\overrightarrow{a}\times \overrightarrow{b}\right|=\left|\overrightarrow{a}.\overrightarrow{b}\right|(\text{Given)} \\ \Rightarrow \left|\overrightarrow{a}\right|\left|\overrightarrow{b}\right|\left|\sin \theta \right|=\left|\overrightarrow{a}\right|\left|\overrightarrow{b}\right|\left|\cos \theta \right| \\ \Rightarrow \left|\sin \theta \right|=\left|\cos \theta \right| \\ \Rightarrow \theta =\frac{\mathrm{\pi }}{4} \end{gathered}$$