pure maths help?

Question

Two non-zero complex number z1 and z2 are such that |z1 + z2| = |z1 - z2. Represent z1,z2,z1 + z2 and z1 - z2 by vectors on an Argand diagram. Hence, or otherwise, find the possible values of aeg(z1/z2).

Answer 1

Let z1 = a + bi and z2 = c + di.

Then, |z1 + z2|^2 = |(a +c) + (b + d)i| = a^2 + 2ac + c^2 + b^2 + 2bd + d^2

And, |z1 - z2|^2 = a^2 - 2ac + c^2 + b^2 - 2bd + d^2

Since |z1 + z2| = |z1 - z2| , it follows 2ac + 2bd = -2ac - 2bd, so that ac + bd =0. On the otgher hand, if ac + bd =0, then |z1 + z2| = |z1 - z2| .

Therefore, ac = -bd. If we think of z1 and z2 as vectors on the Argand Gauss plane, theit dot product is z1 . z2 = ac + bd =0, so that z1 and z2 are perpendicular, supposing none of them is zero.

Since arg(z1/z2) = arg(z1) - arg(z2), and arg(z1) - arg(z2), is either pi/2 or -pi/2, it follows arg(z1/z2) = pi/2 or -pi/2, supposing z1 and z2 are diffrenet from zero.

Answer 2

The two vectors (complex no.s) are at rt angles to each other.
Consider parallelogram (ractangle in this case) z1+z2 is one diagonal and z1-z2 is the other diagonal --- for rectangle, they are both of equal size.

Answer 3

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