4.6 Real three dimensions

When plotting in three dimensions, we need either three Cartesian coordinates (x₀, y₀, z₀) or a magnitude r and two angles φ and θ, as shown in Figure 4.26. coordinates$Cartesian R³

Figure 4.26: Polar coordinates in three dimensions

The magnitude is

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. unit$sphere unit$ball

The unit ball is the set of all points where x₀² + y₀² + z₀² ≤ 1.

Figure 4.27: The unit sphere

We frequently return to the graphic...