3D Coordinate Axes and Planes

Start with an origin and three axes

The origin is the common intersection of the $x$-, $y$– and $z$-axes:

Each axis is a number line with positive and negative directions. The three axes are mutually perpendicular. The $x$-axis consists of points $(x,0,0)$; the $y$-axis of points $(0,y,0)$; and the $z$-axis of points $(0,0,z)$.

The third coordinate is often used for height, but that is a modelling choice, not a requirement of the letter $z$. Before interpreting a real situation, identify what each axis measures. For a room model, you might choose $x$ along one wall, $y$ along the other and $z$ upward, all measured in metres.

OpenStax’s introduction to three-dimensional coordinate systems provides the formal coordinate framework. Its opening material can be studied without calculus.

What right-handed means

Point the fingers of your right hand in the positive $x$ direction and curl them toward positive $y$ through the smaller right angle. Your thumb then points along positive $z$. This fixes the orientation of the coordinate system.

Rotating a drawing of the whole system does not change its handedness. Reversing only one axis does. There is no universal rule that the positive $x$-axis must point toward a particular corner of a page: different views can depict the same right-handed system.

Read an ordered triple in the correct order

The point $P=(2,3,4)$ has $x$-coordinate 2, $y$-coordinate 3 and $z$-coordinate 4. Starting at the origin, one plotting route is to move 2 units in the positive $x$ direction, then 3 units parallel to positive $y$, then 4 units parallel to positive $z$.

The route’s order can change without changing the endpoint, but the order of entries in the coordinate triple cannot. $(2,3,4)$ and $(3,2,4)$ are different points in the same coordinate system.

For $Q=(2,-3,4)$, the second movement is 3 units in the negative $y$ direction. A negative coordinate records direction from a reference plane; it is not a negative distance. The distance from $Q$ to the $xz$-plane is 3 units.

An axis-aligned route is a construction aid, not necessarily the shortest route. For $P=(2,3,4)$, that route has length $2+3+4=9$ units. The direct distance from the origin is $\sqrt{29}$ units, about 5.39. The difference matters when a question distinguishes travel along corridors from straight-line separation.

The three coordinate planes

Each pair of axes determines a plane. In that plane, the coordinate associated with the missing axis is zero:

These are infinite planes, not just the bounded shaded patches used to draw them. The $xy$-plane contains every point $(x,y,0)$, including points with negative $x$ or $y$. It is not limited to the region between the positive axes.

The point $(4,-2,0)$ lies in the $xy$-plane. The point $(0,5,0)$ lies in both the $xy$– and $yz$-planes, so it lies on their intersection, the $y$-axis. The origin lies on all three axes and all three coordinate planes.

A useful classification is: one zero coordinate places a point on at least one coordinate plane; two zero coordinates place it on an axis; three zero coordinates identify the origin. Do not treat these categories as exclusive. An axis is part of two coordinate planes.

Eight octants, with planes as boundaries

The coordinate planes divide the remaining space into eight octants. Each coordinate is positive or negative within an octant, so there are $2^3=8$ possible sign combinations. The positive octant has signs $(+,+,+)$.

For example, $(-2,3,-4)$ lies in the region with signs $(-,+,-)$. Describing an octant by its sign pattern is unambiguous and avoids relying on a numbering convention beyond the positive first octant.

A point with a zero coordinate lies on a coordinate plane, not in the interior of an octant. The zero is not a missing value. It supplies exact information about the point’s location.

Project a point onto a coordinate plane

An orthogonal projection onto a plane follows a perpendicular path to that plane. Projecting onto a coordinate plane sets its perpendicular coordinate to zero and preserves the other two.

For $P=(2,3,4)$:

These are three different points. Projecting onto an axis instead sets two coordinates to zero. The projection of $P$ onto the $x$-axis is $(2,0,0)$, not $(2,3,0)$.

A reflection is different from a projection. Reflecting $P=(2,3,4)$ in the $xy$-plane gives $(2,3,-4)$. The coordinate changes sign rather than disappearing. The reflected point remains 4 units from the plane, on the opposite side; its projection is on the plane.

Distances to planes and axes

In this perpendicular coordinate system, the distance from $(x,y,z)$ to the $xy$-plane is $|z|$. The distances to the $xz$– and $yz$-planes are $|y|$ and $|x|$ respectively. Absolute values convert signed coordinates into non-negative lengths.

For $Q=(2,-3,4)$, the three plane distances are 4, 3 and 2 units. None is negative.

Distance to an axis is different. To reach the $x$-axis, both the $y$ and $z$ components must be removed. The perpendicular distance is

For $Q$, this is $\sqrt{(-3)^2+4^2}=5$ units. Using $|x|=2$ would instead calculate distance to the $yz$-plane. Naming the target before selecting a formula is essential.

Distance between two points

Let $A=(x_1,y_1,z_1)$ and $B=(x_2,y_2,z_2)$. Their coordinate differences are $\Delta x=x_2-x_1$, $\Delta y=y_2-y_1$ and $\Delta z=z_2-z_1$. Apply the Pythagorean theorem to the horizontal displacement, then apply it again with the vertical displacement:

For $A=(1,2,3)$ and $B=(4,6,15)$, the differences are 3, 4 and 12. The displacement parallel to the $xy$-plane has length $\sqrt{3^2+4^2}=5$. The complete distance is

The coordinate-axis route between those points has length $3+4+12=19$ units. The straight segment has length 13 units. Both are valid lengths, but they answer different questions.

If coordinates use different physical units, convert them before applying this Euclidean formula. Squaring an unconverted mixture of centimetres and metres does not produce a meaningful common distance. Likewise, a display can use unequal visual scales along its axes; read the coordinate values rather than measuring its pixels.

Midpoints and displacements

The midpoint of a segment averages corresponding coordinates:

For the points $A$ and $B$ above,

Its distance from each endpoint is 6.5 units, half the full segment length. Negative coordinates are averaged in the same way; do not average their absolute values.

The displacement from $A$ to $B$ is the vector $\langle3,4,12\rangle$. A point describes a location; a vector describes a change of location. Their three entries can look similar, but their roles differ.

For instance, moving from $(2,-3,4)$ by displacement $\langle1,2,-1\rangle$ gives $(3,-1,3)$. Translation changes the coordinates of every moved point by the same amounts while preserving distances between those points.

Equations describe sets, not just isolated points

In three-dimensional space, $x=2$ describes a plane parallel to the $yz$-plane. Both $y$ and $z$ may vary freely. It does not describe only the point $(2,0,0)$.

The simultaneous conditions $x=2$ and $y=-1$ describe a line parallel to the $z$-axis. Adding $z=4$ singles out $(2,-1,4)$. In each case, identify which coordinates are fixed and which remain free.

A more general plane can be written

where $a$, $b$ and $c$ are not all zero. The equation $x+y+z=6$, for example, contains $(1,2,3)$ because $1+2+3=6$. It does not contain $(1,2,4)$. Substitution tests membership without relying on the drawing.

For a plane parallel to a coordinate plane, the equation is simpler. The plane through $(2,-3,4)$ parallel to the $xy$-plane is $z=4$. Every point on it has the same $z$-coordinate, but not the same $x$ and $y$ coordinates.

A circle equation can become a cylinder in space

In a two-dimensional coordinate plane, $x^2+y^2=9$ describes a circle of radius 3 centred at the origin. In three-dimensional space, the same equation imposes no condition on $z$. It therefore describes the infinite cylindrical surface whose horizontal cross-sections are those circles.

To specify just the circle in the $xy$-plane, use both conditions:

A sphere requires all three squared coordinate differences:

For centre $(1,-2,3)$ and radius 5, the equation is $(x-1)^2+(y+2)^2+(z-3)^2=25$. The point $(4,2,3)$ lies on it because the squared differences total $9+16+0=25$. The equation describes the surface. Replacing equality with $\leq$ describes the filled ball, including its boundary.

These examples show why “an equation with $x$ and $y$” is not enough to identify the dimension of its graph. The ambient space and every stated restriction matter.

A final check for a 3D diagram

Start by identifying the origin and positive axis directions. Read a coordinate triple in the stated order. Mark which coordinates are zero, then identify the relevant plane, axis or octant. Use dashed guides to connect a point to its projections without mistaking those guides for edges of a physical object.

Perspective does not preserve every apparent length or angle. The drawn axes may not look perpendicular, and a spatial line can appear to cross another without meeting it. A label or coordinate condition establishes the relationship; the picture helps you follow it.

Related topics

Review Points, Lines and Planes for spatial relationships and Surface Areas and Volumes for measuring the solids that coordinate diagrams represent. Exponents explains the squared quantities in distance and sphere equations.