The 3-D Coordinate System

Main idea

Add a third axis and every point in space gets an address $(x, y, z)$ . The catch is that equations change meaning: one equation now usually describes a surface, not a curve.

Distance

d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2 + (z_2 - z_1)^2}

Sphere

{\htmlData{sym=0}{(x - a)^2}} + (y - b)^2 + (z - c)^2 = {\htmlData{sym=1}{R}}^2

What the symbols mean

$(x - a)^2$
Shift to the center
The center is (a, b, c).
$R$
Radius
Every point on the sphere is this far from the center.

Explanation

The 3D coordinate system is simply a way to give every spot in a room an address: two numbers for where you are on the floor, and a third for how high up you are. It's the setting for everything else in this course, from flight paths to surfaces. The distance formula $\sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2 + (z_2 - z_1)^2}$ looks long, but it's just three differences (how far apart in each direction), squared and added, then square-rooted: Pythagoras with one extra term. Nothing new, just one more coordinate.

You already know the $xy$ -plane. To get to space, stand a third axis, $z$ , straight up out of it. Which way is "up" is a convention, and it's set by the right-hand rule: curl the fingers of your right hand from the positive $x$ -axis to the positive $y$ -axis, and your thumb points along the positive $z$ -axis. Every 3D picture in this course follows it.

The same equation, a new meaning. This is the part that trips people up at first. In the plane, $x = 3$ is a line. In space, $x = 3$ still only says something about $x$ , so $y$ and $z$ can be anything, and you get a whole plane. The same thing happens to $x^2 + y^2 = 4$ . It was a circle, but now $z$ is free, so the circle gets copied at every height and you get a cylinder running along the $z$ -axis. The rule of thumb: a variable that's missing from the equation is free to do whatever it likes.

Spheres come straight from the distance formula, which is the Pythagorean theorem used twice. A sphere is every point at distance $R$ from a center, so squaring the distance formula gives the sphere's equation. If you're handed an expanded equation instead, complete the square in each variable to read off the center and radius, just like you did for circles.

See it

A drone climbing in a spiral. Notice that you need three numbers to say where it is: x and y place its shadow on the ground, and z is how high it sits above that shadow. By the end it's 2 m from the z-axis and about 6.3 m up.

Worked examples

Example 1

Find the center and radius of $x^2 + y^2 + z^2 - 4x + 6y = 3$ .

1
Complete the square in $x$ and in $y$ , adding the same amounts to the right side: $(x^2 - 4x + 4) + (y^2 + 6y + 9) + z^2 = 3 + 4 + 9$ .

Practice

Type your answer and check it. Solve them all to complete the lesson.

Problem 1+5 XP

Find the distance between $(0, 0, 0)$ and $(2, 3, 6)$ .

Problem 2+5 XP

Find the center of the sphere $x^2 + y^2 + z^2 - 2x + 4z = 4$ .

Problem 3+5 XP

Find the radius of that sphere.

Problem 4+5 XP

What is $x^2 + y^2 = 9$ in 3D?

Problem 5+5 XP

Find the midpoint of $(1, 2, 3)$ and $(5, 0, -1)$ .

Problem 6+5 XP

A Wi-Fi router is mounted on the ceiling at $(0, 0, 2.5)$ , and a laptop sits on a desk at $(6, 8, 0.75)$ (meters). How far does the signal travel in a straight line?

Challenge problem+25 XP

The hardest one here. Get it, and you've got this lesson.

Find the point on the $z$ -axis that's equally far from $A(1, 2, 3)$ and $B(2, 0, -1)$ . Give its $z$ -coordinate.

Where it's used

GPS solves for your position as the intersection of spheres around satellites, each of radius equal to that satellite's measured distance.

The Cross Product Equations of Lines