Question

Given that $A=B$ . What is the angle between$\left(\overrightarrow{A}+\overrightarrow{B}\right)$ and $\left(\overrightarrow{A}-\overrightarrow{B}\right)$ ?

Answer 1

$(\overrightarrow{A}+\overrightarrow{B}).(\overrightarrow{A}-\overrightarrow{B})=A^2+\overrightarrow{A}.\overrightarrow{B}-\overrightarrow{A}.\overrightarrow{B}-B^2$

$=A^2-B^2$

given, $A=B$

$\Rightarrow A^2-B^2=0$

$(\overrightarrow{A}+\overrightarrow{B}).(\overrightarrow{A}-\overrightarrow{B})=|A+B|\times |A-B|\times \cos \varphi$

$\Rightarrow 0=|A+B|\times |A-B|\times \cos \varphi$

$\Rightarrow \cos \varphi =0$

$\Rightarrow \varphi ={90}^o$