If Z =a + ib,  Z' = c + id  be two complex numbers such that |Z| = |Z'| = 1  and Re(ZZ'¯) = 0   then ω = a + ic  and ω' = b + id  will be such that

Given |z| =1   ⇒  a2 + b2 = 1   → 1             Given |z1| =1   ⇒  c2 + d2 = 1   → 2             zz1¯ = (a + ib) (c −id)             Given Rc (z z1¯) = 0  ⇒  ac + bd = 0                         ⇒  b =− acd   → 3             From 1, 3,  a2 + a2 c2d2 =1  ⇒   a2 =d2   [∵using 2]             |w| = a2 +c2 = d2 +c2 =1       [∵from 2]             |w1| = b2 +d2 = b2 +a2 =1       [∵from 1]             R|w w1¯| = ab + cd = − a (acd) + cd = 0             |w w1¯| = |w||w1| = 1