Question

under what conditions are u + v and u − v orthogonal? Interpret your result geometrically

Answer 1

Let $\vec u(u_1,u_2,u_3),\vec v(v_1,v_2,v_3)$ then $\vec u-\vec v=\vec a(u_1-v_1,u_2-v_2,u_3-v_3),$

$\vec u+\vec v=\vec b(u_1+v_1,u_2+v_2,u_3+v_3)$ . $\vec a,\vec b$ are ortogonal when the dot product

$\vec a\sdot\vec b=u_1^2-v_1^2+u_2^2-v_2^2+u_3^2-v_3^2=0$ hence $u_1^2+u_2^2+u_3^2=v_1^2+v_2^2+v_3^2$

or $|u|=|v|.$ Next,

$\vec u,\vec v$ are sides of a rhombus, $\vec u+\vec v,\vec u-\vec v$ are diagonals of a rhombus.