Answer to Question #112273 in Analytic Geometry for Noora Almaadeed

Question #112273
under what conditions are u + v and u − v orthogonal? Interpret your result geometrically
1
Expert's answer
2020-04-29T16:20:39-0400

Let u(u1,u2,u3),v(v1,v2,v3)\vec u(u_1,u_2,u_3),\vec v(v_1,v_2,v_3) then uv=a(u1v1,u2v2,u3v3),\vec u-\vec v=\vec a(u_1-v_1,u_2-v_2,u_3-v_3),

u+v=b(u1+v1,u2+v2,u3+v3)\vec u+\vec v=\vec b(u_1+v_1,u_2+v_2,u_3+v_3) . a,b\vec a,\vec b are ortogonal when the dot product

ab=u12v12+u22v22+u32v32=0\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 u12+u22+u32=v12+v22+v32u_1^2+u_2^2+u_3^2=v_1^2+v_2^2+v_3^2

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

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


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