Three-Dimensional Coordinates

In the plane you name a point with two numbers and measure distance with the distance formula, which is the Pythagorean theorem in disguise. Space needs one more number. A point is an ordered triple $(x, y, z)$: go $x$ units along the $x$-axis, $y$ units parallel to the $y$-axis, then $z$ units up (or down, if $z$ is negative).

The three axes are perpendicular, and their directions 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 along the positive $z$-axis. Every textbook picture and every formula later in this course (the cross product, orientations, Stokes' theorem) assumes this arrangement.

Key idea

A point in space is $(x, y, z)$. The distance between $(x_1, y_1, z_1)$ and $(x_2, y_2, z_2)$ is $$d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2 + (z_2 - z_1)^2}.$$ The sphere with center $(a, b, c)$ and radius $r$ is every point at distance $r$ from the center: $(x - a)^2 + (y - b)^2 + (z - c)^2 = r^2$.

Why it works

Use the Pythagorean theorem twice. Walk from $P$ to $Q$ in three legs: $\Delta x$ along the $x$-direction, $\Delta y$ along the $y$-direction, then $\Delta z$ straight up. The first two legs lie in a horizontal floor and are perpendicular, so the floor diagonal has length $\sqrt{(\Delta x)^2 + (\Delta y)^2}$. That diagonal is horizontal and the last leg is vertical, so they are perpendicular too, and $$d^2 = \left(\sqrt{(\Delta x)^2 + (\Delta y)^2}\right)^2 + (\Delta z)^2 = (\Delta x)^2 + (\Delta y)^2 + (\Delta z)^2.$$

The three colored legs are $\Delta x$, $\Delta y$ and $\Delta z$. Drag the picture to turn it: the legs always meet at right angles. The dashed box under $P$ shows how its coordinates are measured.

In space, one equation usually describes a surface, not a curve.
Equation	The set of points
$z = 0$	the $xy$-plane (the floor)
$x = 0$	the $yz$-plane
$y = 0$	the $xz$-plane
$x = 3$	a plane parallel to the $yz$-plane, 3 units in front of it
$x = 0$ and $y = 0$	the $z$-axis

Worked examples

Worked example: Distance

Find the distance between $P(1, -2, 3)$ and $Q(4, 2, 15)$.

Differences: $\Delta x = 3$, $\Delta y = 4$, $\Delta z = 12$.
Square and add: ${}9 + 16 + 144 = 169$.
$d = \sqrt{169} = 13$.

Answer: ${}13$

Worked example: Center and radius of a sphere

Find the center and radius of $x^2 + y^2 + z^2 - 4x + 6z - 3 = 0$.

Group by variable: $(x^2 - 4x) + y^2 + (z^2 + 6z) = 3$.
Complete each square: $x^2 - 4x = (x - 2)^2 - 4$ and $z^2 + 6z = (z + 3)^2 - 9$.
So $$(x - 2)^2 + y^2 + (z + 3)^2 = 3 + 4 + 9 = 16.$$
Center $(2, 0, -3)$ (the signs flip), radius $\sqrt{16} = 4$.

Answer: center $(2, 0, -3)$, radius ${}4$

Worked example: Distance to an axis and to a plane

How far is $(3, -4, 5)$ from the $z$-axis, and from the $xy$-plane?

The nearest point of the $z$-axis is $(0, 0, 5)$: same height, $x = y = 0$.
So the distance to the axis is $\sqrt{3^2 + (-4)^2} = 5$: only $x$ and $y$ count.
The nearest point of the $xy$-plane is $(3, -4, 0)$, straight below, so that distance is $|z| = 5$.

Answer: ${}5$ and ${}5$

Common mistake

Dropping the $z$-term. Using $\sqrt{(\Delta x)^2 + (\Delta y)^2}$ in space gives the length of the shadow on the floor, not the distance. Check: the answer can never be smaller than $|\Delta z|$.
Left-handed axes. If your sketch has $x$ to the right, $y$ up the page and $z$ coming out of the page toward you, that is fine; but $z$ pointing into the page makes a left-handed system and later sign rules will fail. Test with the right-hand rule: curl from $x$ to $y$, and the thumb must point along $z$.

Three-Dimensional Coordinates

Worked examples

Practice

Builds on

Leads to

Sources and further reading