$\begin{array}{l}
Let\,the\,two\,vectors\,be\,\vec A\,and\,\vec B\,.\\
Then,\,magnitude\,of\,sum\,of\,\vec A\,and\,\vec B\,,\\
\,\,\,\,\,\,\,\,\left| {\vec A + \vec B} \right| = \sqrt {{A^2} + {B^2} + 2AB\cos \theta } \\
and\,magnitude\,od\,difference\,of\,\vec A\,and\,\vec B\,,\\
\,\,\,\,\,\,\,\left| {\vec A – \vec B} \right| = \sqrt {{A^2} + {B^2} – 2AB\cos \theta } \,,\\
\,\,\,\,\,\,\,\left| {\vec A + \vec B} \right|\, = \,\,\left| {\vec A – \vec B} \right|\,\,\left( {given} \right)
\end{array}$

$\begin{array}{l}
or\,\,\sqrt {{A^2} + {B^2} + 2AB\cos \theta } \\
\,\,\,\, = \sqrt {{A^2} + {B^2} – 2AB\cos \theta } \\
 \Rightarrow \,\,4\,AB\cos \theta  = 0\\
\,\,\,\,4\,AB \ne 0,\,\,\therefore \cos \theta  = 0\,or\,\theta  = {90^ \circ }
\end{array}$