Mathematics - Vector Question with Solution | TestHub

$\left|\overrightarrow{a}\right|=\left|\overrightarrow{b}\right|,|\overrightarrow{a}\times \overrightarrow{b}|=\left|\overrightarrow{a}\right|,\overrightarrow{a}\perp \overrightarrow{b}$

$|\overrightarrow{a}\times \overrightarrow{b}|=\left|\overrightarrow{a}\right|\Rightarrow \left|\overrightarrow{a}\right|\left|\overrightarrow{b}\right|\sin 90°=\left|\overrightarrow{a}\right|\Rightarrow \left|\overrightarrow{b}\right|=1=\left|\overrightarrow{a}\right|$

$\overrightarrow{a}$ and $\overrightarrow{b}$ are mutually perpendicular unit vectors.

Let $\overrightarrow{a}=\hat{i},\overrightarrow{b}=\hat{j}\Rightarrow \overrightarrow{a}\times \overrightarrow{b}=\hat{k}$

$\cos \theta =\frac{(\hat{i}+\hat{j}+\hat{k})\hat{i}}{\sqrt{3}\sqrt{1}}=\frac{1}{\sqrt{3}}\Rightarrow \theta ={\cos }^{-1}\left(\frac{1}{\sqrt{3}}\right)$