Question

$\mathrm{If}\left|\overrightarrow{a}\right|=2,\left|\overrightarrow{b}\right|=5\mathrm{and}\left|\overrightarrow{a}\times \overrightarrow{b}\right|=8,\mathrm{find}\overrightarrow{a}·\overrightarrow{b}.$

Answer 1

$$\begin{gathered} \mathrm{We}\mathrm{know} \\ {\left(\overrightarrow{a}.\overrightarrow{b}\right)}^2+{\left|\overrightarrow{a}\times \overrightarrow{b}\right|}^2={\left|\overrightarrow{a}\right|}^2{\left|\overrightarrow{b}\right|}^2 \\ \Rightarrow {\left(\overrightarrow{a}.\overrightarrow{b}\right)}^2+8^2=2^2\times 5^2(\because \left|\overrightarrow{a}\times \overrightarrow{b}\right|=8,\left|\overrightarrow{a}\right|=2\mathrm{and}\left|\overrightarrow{b}\right|=5) \\ \Rightarrow {\left(\overrightarrow{a}.\overrightarrow{b}\right)}^2+64=100 \\ \Rightarrow {\left(\overrightarrow{a}.\overrightarrow{b}\right)}^2=36 \\ \Rightarrow \left(\overrightarrow{a}.\overrightarrow{b}\right)=6 \end{gathered}$$