Non-Euclidean Geometry Homework

The following homework is based on a complex number unit we are completing in my Non-Euclidean class.

Problem: Complex numbers and (plane) vectors provide complementary descriptions of the Euclidean plane once we identify z with the vector (real number(z);imaginary(z)), and conversely, the vector (x; y) with the complex number x + iy.

(a) Show that realnumber (z*mod(w)) is the dot product of two vectors z = x + iy and w = u + iv.

(b) Use (a) to show that `*l*0z and `*l*0w are perpendicular if and only if w = tiz for some real number t.

(c) Verify (b) another way by expressing t*i* in polar form and interpreting w = tiz as a rotation.

If you can help, that would be greatly appreciated.