# two vectors a and b are of equal lengths and mutually perpendicular. show that their sum and difference will be of same length and mutually perpendicular.

$\\\vec a,\vec b \\\vec a.\vec b=0 \\\vec v_1=\vec a+\vec b \\\vec v_2=\vec a-\vec b \\|\vec v_1|=|\vec a|^2+|\vec b|^2+2*|\vec a||\vec b|*0 \\|\vec v_2|=|\vec a|^2+|\vec b|^2-2*|\vec a||\vec b|*0 \\=>|\vec v_1|=|\vec v_2| \\=>v_1.\vec v_2=(\vec a+\vec b).(\vec a-\vec b) \\=>v_1.\vec v_2=|\vec a|^2-\vec a .\vec b+\vec a.\vec b-|\vec b|^2=0$