Question

If $\overrightarrow{b}$is a unit vector such that $\left(\overrightarrow{a}+\overrightarrow{b}\right).\left(\overrightarrow{a}-\overrightarrow{b}\right)=8,\mathrm{find}\left|\overrightarrow{a}\right|.$

Answer 1

$$\begin{gathered} \text{Given that}\overrightarrow{b}\text{is a unit vector.} \\ \therefore \left|\overrightarrow{b}\right|=1 \\ \mathrm{And} \\ \left(\overrightarrow{a}+\overrightarrow{b}\right).\left(\overrightarrow{a}-\overrightarrow{b}\right)=8\left(\mathrm{Given}\right) \\ \Rightarrow {\left|\overrightarrow{a}\right|}^2-{\left|\overrightarrow{b}\right|}^2=8 \\ \Rightarrow {\left|\overrightarrow{a}\right|}^2-1^2=8 \\ \Rightarrow {\left|\overrightarrow{a}\right|}^2=9 \\ \therefore \left|\overrightarrow{a}\right|=3 \end{gathered}$$