4.6 Real three dimensions
When plotting in three dimensions, we need either three Cartesian coordinates (x0, y0, z0) or a radius r and two angles ϕ and θ.
The radius r = |P| = √x02 + y02 + z02. ϕ is the angle from the positive x axis to the dotted line from (0, 0) to the projection (x0, y0) of P into the xy-plane. θ is the angle from the positive z axis to the line segment 0P.
That’s a lot to absorb, but it builds up systematically from what we saw in R2. When r = 1 we get the unit sphere in R3. It’s the set of all points (x0, y0, z0) in R3 where x02 + y02 + z02 = 1. The unit ball is the set of all points where x02 + y02 + z02 ≤ 1.
We return to this graphic frequently when we consider the Bloch sphere representation of a qubit.
If we were in four real dimensions we would need 4 coordinates (x, y, z, w). The unit hypersphere is the set of all points such that...
