Question

The vector sum of two forces is perpendicular to their vector differences.

Prove that the two vector forces are equal in magnitude. Help !

Answer 1

$$\begin{gathered} \left(\overrightarrow{A}+\overrightarrow{B}\right).\left(\overrightarrow{A}-\overrightarrow{B}\right)=0 \\ \Rightarrow \overrightarrow{A}.\overrightarrow{A}+\overrightarrow{B}.\overrightarrow{A}-\overrightarrow{A}.\overrightarrow{B}-\overrightarrow{B}.\overrightarrow{B}=0 \\ \Rightarrow {\left|\overrightarrow{A}\right|}^2={\left|\overrightarrow{B}\right|}^2\Rightarrow \left|\overrightarrow{A}\right|=\left|\overrightarrow{B}\right| \end{gathered}$$