Chapter 8 : Coordinate Geometry

Study Questions and Projects

What are geometric significants of complex number addition, scalar multiplication, and complex conjugate? What is the geometric representation of complex number multiplication?
Complex numbers are used to describe surfaces in Euclidean 4-space. We can find formulas for ordinary functions of two variables x and y by expanding algebraic expressions involiving the imaginary unit i, for example z^2 = (x+iy)^2 = (x^2 - y^2) + 2xyi.
a) Find the comparable expressions for z^3 = (x+iy)^3 and z^4 = (x+iy)^4 . What patterns emerge?
b) Find expressions for 1/z and 1/z^2 using the fact that z = (x+iy) = x-iy. and zz = x^2 + y^2.
What are n-tuple coordinates of the unit n-dimensional hypercube similar to 2,3, and 4 dimensional cases in "Coordinate Geometry"? How many vertices does it have? How coordinate system make counting vertices easier?
In 2d, we can find a relation to change (x,y) in Cartesian Coordinate, to (r,θ) in Polar Coordinate. Also, a point (x,y,z) in 3d can be changed to Spherical Coordinate as well. Base on what you learn from "Coordinate for Circles and Spheres," Derive equations that change (x,y,u,v) in 4-dimensional Cartesian coordinate to a coordinate in 4-sphere system. What should be a good parameters in that system? How about a point (x1,x2,...,xn) in n-dimensional Cartesian Coordinate to a representation in n-sphere coordinate system?