Question

If $\left|\overrightarrow{a}+\overrightarrow{b}\right|=60,\left|\overrightarrow{a}-\overrightarrow{b}\right|=40\mathrm{and}\left|\overrightarrow{b}\right|=46,\mathrm{find}\left|\overrightarrow{a}\right|$

Answer 1

$$\begin{gathered} \text{We know that} \\ {\left|\overrightarrow{a}+\overrightarrow{b}\right|}^2+{\left|\overrightarrow{a}-\overrightarrow{b}\right|}^2=2\left({\left|\overrightarrow{a}\right|}^2+{\left|\overrightarrow{b}\right|}^2\right) \\ \Rightarrow {60}^2+{40}^2=2\left({\left|\overrightarrow{a}\right|}^2+{46}^2\right)\left(\mathrm{Given}\right) \\ \Rightarrow 3600+1600=2{\left|\overrightarrow{a}\right|}^2+4232 \\ \Rightarrow 968=2{\left|\overrightarrow{a}\right|}^2 \\ \Rightarrow {\left|\overrightarrow{a}\right|}^2=484 \\ \Rightarrow \left|\overrightarrow{a}\right|=22 \end{gathered}$$