Solve the problem.The unit vectors u and v are combined to produce two new vectors and Show that a and b are orthogonal. Assume

Question

What will be an ideal response?

Answer 1

u = u_x i + u_yj and v = v_x i + v_yj , so
a = u + v = (u_x + v_x) i + (u_y + v_y)j and b = u - v = (u_x - v_x) i + (u_y - v_y)j

Take the dot product a ? b:
a ? b = (u + v) ? (u - v) = (u_x + v_x)(u_x - v_x) + (u_y + v_y)(u_y - v_y)
 =  -  +  -   = ( + ) - (  + )
 =  -   = 1 - 1 = 0
Since the dot product of the two non-zero vectors is zero they are orthogonal.