Question

$\mathrm{If}\left|\overrightarrow{a}\right|=\sqrt{26},\left|\overrightarrow{b}\right|=7\mathrm{and}\left|\overrightarrow{a}\times \overrightarrow{b}\right|=35,\mathrm{find}\overrightarrow{a}.\overrightarrow{b}.$

Answer 1

$$\begin{gathered} \mathrm{We}\mathrm{know} \\ {\left(\overrightarrow{a}.\overrightarrow{b}\right)}^2+{\left|\overrightarrow{a}\times \overrightarrow{b}\right|}^2={\left|\overrightarrow{a}\right|}^2{\left|\overrightarrow{b}\right|}^2 \\ \Rightarrow {\left(\overrightarrow{a}.\overrightarrow{b}\right)}^2+{35}^2={\left(\sqrt{26}\right)}^2\times 7^2(\because \left|\overrightarrow{a}\times \overrightarrow{b}\right|=35,\left|\overrightarrow{a}\right|=\sqrt{26}\mathrm{and}\left|\overrightarrow{b}\right|=7) \\ \Rightarrow {\left(\overrightarrow{a}.\overrightarrow{b}\right)}^2+1225=1274 \\ \Rightarrow {\left(\overrightarrow{a}.\overrightarrow{b}\right)}^2=49 \\ \Rightarrow \left(\overrightarrow{a}.\overrightarrow{b}\right)=7 \end{gathered}$$