Coordinates in space and octants

In two dimensions, every point has two coordinates $(x, y)$. In three dimensions, we add a third axis perpendicular to both , the $z$-axis. Every point is then named by three numbers $(x, y, z)$.

The three coordinate axes

The three axes , $x$, $y$, $z$ , are mutually perpendicular and meet at the origin $O = (0, 0, 0)$. By convention we use a right-handed coordinate system: if you curl your right hand from the positive $x$-axis to the positive $y$-axis, your thumb points along the positive $z$-axis.

Coordinate planes

Pairs of axes determine three coordinate planes:

• $xy$-plane: $z = 0$. The "floor" , points with $z = 0$.
• $yz$-plane: $x = 0$. The wall facing you.
• $zx$-plane (or $xz$-plane): $y = 0$.

A point is on the $xy$-plane iff its $z$-coordinate is $0$. Similarly for the others.

Octants

The three coordinate planes divide space into eight regions called octants. Octants are labelled by the signs of the three coordinates. The conventional first octant is $(+, +, +)$ , all coordinates positive. The others are listed below:

(Different textbooks may number them differently; the signs are universal.)

Locating points

To plot $(3, 4, 5)$: go $3$ units along the positive $x$-axis, then $4$ along positive $y$, then $5$ up along positive $z$. The point lies in the first octant.

A point of the form $(a, 0, 0)$ lies on the $x$-axis; $(0, b, 0)$ on the $y$-axis; $(0, 0, c)$ on the $z$-axis. A point on the $xy$-plane has $z = 0$.

Distance from coordinate planes and axes

• Distance from $(x_0, y_0, z_0)$ to the $xy$-plane is $|z_0|$.
• Distance to the $yz$-plane is $|x_0|$.
• Distance to the $zx$-plane is $|y_0|$.
• Distance from the $x$-axis is $\sqrt{y_0^2 + z_0^2}$.
• Distance from the $y$-axis is $\sqrt{x_0^2 + z_0^2}$.
• Distance from the $z$-axis is $\sqrt{x_0^2 + y_0^2}$.

Worked examples

Example 1. Identify the octant of $(2, -3, 5)$.

Signs $(+, -, +)$ , octant IV.

Example 2. Find the distance from $(2, -3, 5)$ to (a) the $xy$-plane, (b) the $x$-axis.

(a) $|5| = 5$. (b) $\sqrt{9 + 25} = \sqrt{34}$.

Example 3. What is the equation of the $xy$-plane?

$z = 0$.

Example 4. Describe the set of points with $y = 2$.

A plane parallel to the $zx$-plane, $2$ units along the positive $y$-axis.

Example 5. A point lies on the $x$-axis and is at distance $5$ from the origin. Find its coordinates.

$(\pm 5, 0, 0)$ , two possibilities.

Try it yourself

1. Identify the octant: $(-1, -2, 3)$.
2. Identify the octant: $(4, 5, -6)$.
3. Plot the points $(0, 0, 3)$ and $(2, 0, 0)$.
4. Distance from $(3, -4, 5)$ to the $xy$-plane.
5. Distance from $(3, -4, 5)$ to the $y$-axis.
6. The set of points equidistant from origin in 3D: describe.
7. The set of points with $z = 4$: describe.
8. Coordinates of a point on the $y$-axis at distance $7$ from origin.
9. Describe the set $\{(x, y, z) : x = y\}$ in space.
10. A point is in the second octant ($x < 0, y > 0, z > 0$) and has coordinates $(-3, 4, ?)$. Give one possible value for $?$.
11. A point on the $zx$-plane has coordinates $(2, ?, 5)$. What is $?$?
12. Identify the location: $(0, 3, 0)$.

Pitfalls / Tricks

• The $z$-coordinate measures height off the $xy$-plane , easy to forget the sign.
• Points on a coordinate axis have two zero coordinates; points on a coordinate plane have one zero coordinate.
• Distance from an axis is not the distance from a single coordinate , it's the Pythagorean combination of the other two.
• Insight. The right-hand rule fixes the orientation. Always sketch axes consistently , flipping the $z$-axis flips your intuition.