TION 5 3 points Let A B C be three points in the complex pla

TION 5. (3 points) Let A, B, C be three points in the complex plane corresponding to the QUES complex numbers a, b, c E C. Assume that lal-b|-\\c] = 1. Also, let P be the point corresponding to a + b+c. Prove that the line through P A is perpendicular to Bc. Solution:

Solution

Consider - |a| = |b| = |c| = 1

a, b and c are complex numbers on complex plane.

So a, b, c will be on a circle with radius one.

Let for simplification -

A = a = (1/2,1/2), B = b = (-1/2,1/2), C = c = (0,-1)

P = a+b+c = (0,2 - 1)

PA = (1/2, 1/2 - (2 - 1)) = (1/2, (2-1)/2)

BC = (1/2, -1-1/2)

So, PA•BC = 0

PA•BC = 1/2×1/2 + (1-1/2)×(-1-1/2)

PA•BC = 1/2 -1 -1/2+1/2+1/2

PA•BC = 0

Hence PA and BC are perpendicular.