Question

Two vectors $\vec{a}\ and\ \vec{b}$ are such that $|\vec{a}+\vec{b}|=|\vec{a}-\vec{b}|$. What is the angle between $\vec{a}\ and\ \vec{b}$$?$

Answer 1

$</p><p>\tt{\huge{\pink{Hello!!! }}}$

$\tt{ \orange{ \phi \: = \: 90 °\: }}$

Answer 2

$\begin{gathered}\large {\boxed{\sf{\mid{\overline {\underline {\star ANSWER ::}}}\mid}}}\end{gathered}$

Given :-

$\bold{➪ \: |\vec{a}+\vec{b}|=|\vec{a}-\vec{b}|}$

To Find :-

$\bold{➪ \: Angle \: between \: \vec{a}\ and\ \vec{b}}$

Solution :-

$|\vec{a}+\vec{b}|=|\vec{a}-\vec{b}| \\ \\ \dag \: \underline {Squaring \: on \: both \: sides : } \\ |\vec{a}+\vec{b}| {}^{2} =|\vec{a}-\vec{b}| {}^{2} \\ \\ |\vec{a}| {}^{2} + |\vec{b}| {}^{2} + 2 \: \vec{a} \: \vec{b} = |\vec{a}| {}^{2} + |\vec{b}| {}^{2} - 2 \: \vec{a} \: \vec{b} \\ \\ 2 \: \vec{a} \: \vec{b} = - 2 \: \vec{a} \: \vec{b} \\ \\ 4 \: \vec{a} \: \vec{b} = 0 \\ \\ \vec{a} \: \vec{b} = 0 \\ \\ \large \boxed{ \bold{\therefore \: \vec{a} \: is \: perpendicular \: to \: \vec{b}}} \\ \\ \large \boxed{ \bold{\star \: \theta = 90 \degree}}$

@SweetestBitter