Calc 3 Notes - Stewart - Chapter 12.1: Three-Dimensional Coordinate Systems

This is an entry in a series of notes following my learning of the James Stewart - Calculus Early Transcendentals 7e, chapters 12-16.

Prior to Calculus 3, the focus of math has ultimately been the study of manipulating 1 and 2 dimensional objects sketched on a 2d plane. At the very end of Calculus 2, learning how to create 3D objects by rotating a curve around either the x or y axis gave a hint of things to come.

To review, a 2d world is divided up into a grid consisting of axis that represent the left and right directions, often called the x-axis, and the up and down directions, the y-axis. Any location on the plane can be described by how far left or right, and up or down, to go. This location or address in the 2d world is known as point and is represented by an ordered pair, (x, y). The two axis meet at a location known as the origin, or (0, 0). The x and y axis divide up the grid into 4 quadrants and intersect and the point (0, 0), also known as the origin. The diagram below shows the point (3,4) plotted in 2d space. Piece of cake, right?

Now enter the 3d world. What does that mean? It means there is a new direction, perpendicular to the two directions available in 2d space, commonly referred to as the z-axis. At the fundamental level, describing 3d space is the same as describing 2d space, only this new extra axis is included. A point in the 3d space now takes 3 coordinates to describe it. That is, to get to a point in 3d space from the origin, one must travel a certain distance along the x-axis, then a certain distance on the y-axis, and finally a certain distance on the z-axis. Intuitively, the origin for the 3d space is (0, 0, 0).

Drawing diagrams of 3d space can be more challenging than that of the 2d space. Because a 3d space is divided into 8 sections, it can be quite a bear to draw, and so focus is generally given to the first octant, where all values are positive, like the following diagram.

The first thing one might notice looking at an empty, labeled, 3d grid-space, is that the old familiar 2d world is right there. The x,y plane is, in the 3d space, a collection of all the points that exist with a 0 for the z coordinate. The x,z and y,z planes are also 2d worlds that exists in the 3d space, with y=0 and x=0 for the 3rd coordinate respectively.