Three-Dimensional Coordinates
Textbook Reference
| Primary source | OpenStax Calculus Volume 3, Section 2.1: “Vectors in the Plane” and Section 1.1: “Parametric Equations” |
| Supplementary | OpenStax Calculus Volume 3, Section 2.2: “Vectors in Three Dimensions” |
| Supplementary link | https://openstax.org/books/calculus-volume-3/pages/2-2-vectors-in-three-dimensions |
| Textbook used in class | Stewart, Calculus, Section 12.1: “Three-Dimensional Coordinate Systems” (Examples 1, 2) |
Opening Scenario
A GPS unit reports your position as latitude, longitude, and altitude: three numbers that locate you in space. The three-dimensional coordinate system is the mathematical version of that idea. Every point in space corresponds to an ordered triple $(x, y, z)$, and two-dimensional geometry -- lines, planes, distances -- extends cleanly into three dimensions.
Quick Reference
The $xyz$-coordinate system. Three mutually perpendicular axes meet at the origin $O = (0, 0, 0)$:
- The $x$-axis and $y$-axis lie in the horizontal plane.
- The $z$-axis is vertical.
- The axes follow the right-hand rule: point the fingers of your right hand along the positive $x$-axis and curl them toward the positive $y$-axis; your thumb points in the positive $z$-direction.
Coordinate planes:
- $xy$-plane: $z = 0$
- $xz$-plane: $y = 0$
- $yz$-plane: $x = 0$
These three planes divide space into eight octants. The first octant has $x > 0$, $y > 0$, $z > 0$.
Planes parallel to coordinate planes: The equation $z = c$ (constant) describes a horizontal plane at height $c$, parallel to the $xy$-plane. Similarly $x = c$ is a plane parallel to the $yz$-plane, and $y = c$ is parallel to the $xz$-plane.
Key Concepts
1. Plotting Points in 3D
To plot $(x_0, y_0, z_0)$: start at the origin, move $x_0$ units along the $x$-axis, then $y_0$ units parallel to the $y$-axis, then $z_0$ units parallel to the $z$-axis.
Example 1. Locate the points $P = (3, -2, 4)$ and $Q = (-1, 0, 2)$. (Stewart 12.1, Example 1.)
For $P$: move $3$ right along $x$, $2$ back along $y$ (since $-2$), and $4$ up along $z$. For $Q$: move $1$ left along $x$, stay at the $xz$-plane ($y = 0$), and $2$ up.
Recap. The coordinate $y = 0$ for $Q$ means the point lies in the $xz$-plane. Points with one coordinate equal to zero lie in one of the three coordinate planes.
2. Equations Describing Simple Regions
A single equation in three variables describes a surface (not a curve).
- $z = 3$: all points at height $3$, a horizontal plane.
- $x = -1$: all points with $x$-coordinate $-1$, a vertical plane.
- $z \geq 0$: the closed upper half-space (at or above the $xy$-plane).
Example 2. What does the equation $y = 4$ describe in 3D? (Stewart 12.1, Example 2.)
In 3D, $y = 4$ is a vertical plane parallel to the $xz$-plane, containing all points $(x, 4, z)$ for any $x$ and $z$.
Boxed answer: A plane parallel to the $xz$-plane, at $y = 4$.
Recap. In 2D the equation $y = 4$ is a horizontal line. In 3D it becomes a plane -- the two-dimensional constraint releases the third coordinate completely, so all values of $x$ and $z$ are allowed.
confusing 2D and 3D interpretations of an equation. The equation $x^2 + y^2 = 4$ describes a circle in the $xy$-plane. In 3D it describes an infinite cylinder of radius $2$ centered on the $z$-axis, because $z$ is unrestricted. The dimension of the space matters for every equation you encounter.
Common Errors Summary
| Error | Correction |
|---|---|
| Confusing $x$-axis and $y$-axis orientation | Use the right-hand rule; the positive $z$-axis points up if $x$ is to the right and $y$ is forward |
| Reading $z = c$ as the $z$-axis | $z = c$ is a plane, not a line; the $z$-axis is the set of all points $(0, 0, z)$ |
| Thinking a one-variable equation in 3D gives a line | One equation gives a surface; you need two equations for a line in 3D |
Common Misconceptions
an equation in three variables always describes a curve.
This is the iconic-graph error. In two dimensions, one equation such as $x^2 + y^2 = 4$ describes a curve (a circle). In three dimensions, one equation describes a surface, because the missing variable is unconstrained. The equation $x^2 + y^2 = 4$ in 3D is an infinite cylinder of radius 2 centered on the $z$-axis. A curve in 3D requires two simultaneous equations.
Leveled Practice
Problem 1. Describe the set of all points $(x, y, z)$ satisfying $x^2 + z^2 = 9$, $y = 2$.
Show answer
$y = 2$ restricts to the plane at height $y = 2$. In that plane, $x^2 + z^2 = 9$ is a circle of radius $3$. Together: a circle of radius $3$ in the plane $y = 2$.
Problem 2. Describe the set $\{(x,y,z) : x^2 + y^2 \leq 1,\; 0 \leq z \leq 2\}$.
Show answer
$x^2 + y^2 \leq 1$ is a disk of radius $1$ in the $xy$-plane. $0 \leq z \leq 2$ stacks that disk from height $0$ to $2$. Together: a solid cylinder of radius $1$ and height $2$ with its base in the $xy$-plane.