Question

For any two vectors $\overrightarrow{a}\mathrm{and}\overrightarrow{b}$write the value of ${\left(\overrightarrow{a}.\overrightarrow{b}\right)}^2+{\left|\overrightarrow{a}\times \overrightarrow{b}\right|}^2$ in terms of their magnitudes.

Answer 1

$$\begin{gathered} {\left(\overrightarrow{a}.\overrightarrow{b}\right)}^2+{\left|\overrightarrow{a}\times \overrightarrow{b}\right|}^2 \\ ={\left(\left|\overrightarrow{a}\right|\left|\overrightarrow{b}\right|\cos \theta \right)}^2+{\left(\left|\overrightarrow{a}\right|\left|\overrightarrow{b}\right|\sin \theta \right)}^2 \\ ={\left|\overrightarrow{a}\right|}^2{\left|\overrightarrow{b}\right|}^2\left({\cos }^2\theta +{\sin }^2\theta \right) \\ ={\left|\overrightarrow{a}\right|}^2{\left|\overrightarrow{b}\right|}^2\left(1\right) \\ ={\left|\overrightarrow{a}\right|}^2{\left|\overrightarrow{b}\right|}^2 \end{gathered}$$