Three-Dimensional Coordinate Geometry

Introduction

students, imagine standing in a room and pointing to a fly on the ceiling. In two dimensions, you can describe a point on a flat map using two numbers. But in real life, objects and positions often need a third number too. That is the main idea of three-dimensional coordinate geometry ✨. It gives us a way to describe points, lines, and distances in space using coordinates, just like in a plane, but with one extra direction.

In this lesson, you will learn how to:

describe points in

\mathbb{R}^3

using coordinates like $(x,y,z)$,

calculate distances and midpoints in three-dimensional space,
understand vectors and lines in space,
use equations to represent planes,
connect these ideas to the wider IB topic of Geometry and Trigonometry.

These ideas are useful in physics, architecture, engineering, computer graphics, and navigation 📍. By the end, students, you should be able to explain the key language and use standard IB procedures for three-dimensional coordinate geometry.

Points, axes, and distance in space

Three-dimensional coordinate geometry uses three perpendicular axes: the $x$-axis, the $y$-axis, and the $z$-axis. These meet at the origin, $(0,0,0)$. A point is written as $(x,y,z)$, where each coordinate tells you how far to move along one axis.

For example, the point $(3,-2,5)$ means:

move $3$ units in the positive $x$-direction,
move $2$ units in the negative $y$-direction,
move $5$ units in the positive $z$-direction.

This is like finding a seat in a 3D auditorium: row, column, and level all matter 🎭.

The distance between two points $A(x_1,y_1,z_1)$ and $B(x_2,y_2,z_2)$ comes from the three-dimensional version of Pythagoras’ theorem:

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

This formula is essential because it tells us the straight-line distance through space.

Example: find the distance between $A(1,2,3)$ and $B(4,6,3)$.

Substitute into the formula:

$AB=\sqrt{(4-1)^2+(6-2)^2+(3-3)^2}$

$=\sqrt{3^2+4^2+0^2}$

$=5.$

So the distance is $5$ units.

A midpoint is the point halfway between two points. If $A(x_1,y_1,z_1)$ and $B(x_2,y_2,z_2)$, then the midpoint $M$ is

$M\left(\frac{x_1+x_2}{2},\frac{y_1+y_2}{2},\frac{z_1+z_2}{2}\right).$

This is useful when finding the center of a segment or checking symmetry.

Vectors and lines in three dimensions

Vectors are an important part of coordinate geometry because they describe direction and size. A vector in space can be written as

$\mathbf{a}=\begin{pmatrix}a_1\a_2\a_3\end{pmatrix}.$

The vector from $A(x_1,y_1,z_1)$ to $B(x_2,y_2,z_2)$ is

$\overrightarrow{AB}=\begin{pmatrix}x_2-x_1\y_2-y_1\z_2-z_1\end{pmatrix}.$

This vector tells you the move needed to go from $A$ to $B$.

A line in space can be written in vector form as

$\mathbf{r}=\mathbf{a}+t\mathbf{d},$

where $\mathbf{a}$ is a position vector of a point on the line, $\mathbf{d}$ is a direction vector, and $t$ is a parameter.

In component form, a line can be written as

$\begin{pmatrix}$x\y\z$\end{pmatrix}$=$\begin{pmatrix}$x_0\y_0\z_0$\end{pmatrix}$+t$\begin{pmatrix}$a\b\c$\end{pmatrix}$.

This means the coordinates change at a constant rate as $t$ changes.

Example: a line through $(1,0,2)$ with direction vector $\begin{pmatrix}2\\-1\\3\end{pmatrix}$ has equation

$\begin{pmatrix}$x\y\z$\end{pmatrix}$=$\begin{pmatrix}1$\\0\\$2\end{pmatrix}$+t$\begin{pmatrix}2$\\-1\\$3\end{pmatrix}$.

So the coordinates are

$x=1+2t,\quad y=-t,\quad z=2+3t.$

A common task is checking whether a point lies on a line. If a point fits the same parameter $t$ in all three coordinate equations, then it lies on the line. This reasoning is widely used in IB questions.

Another useful idea is the angle between two vectors. If vectors $\mathbf{u}$ and $\mathbf{v}$ have dot product

$\mathbf{u}\cdot\mathbf{v}=|\mathbf{u}||\mathbf{v}|\cos\theta,$

then the angle $\theta$ between them can be found. This helps in deciding whether lines are parallel or perpendicular.

Planes in space

A plane is a flat surface extending in two dimensions inside three-dimensional space. Think of a tabletop or a wall 🧱. In coordinate geometry, planes are often described by equations such as

$ax+by+cz=d,$

where $a$, $b$, and $c$ are not all zero.

This form is very important because it shows all points $(x,y,z)$ that satisfy the same rule. For example, the plane

$2x-y+z=5$

contains every point whose coordinates make the equation true.

The vector $\begin{pmatrix}a\b\c\end{pmatrix}$ is normal to the plane, meaning it is perpendicular to the plane. This is useful because if a line is perpendicular to a plane, its direction vector is parallel to the normal vector.

Example: if a plane has equation

$3x+4y-2z=7,$

then a normal vector is

$\begin{pmatrix}3\\4\\-2\end{pmatrix}.$

So a line perpendicular to this plane could have direction vector

$\begin{pmatrix}3\\4\\-2\end{pmatrix}.$

To check whether a point lies in a plane, substitute its coordinates into the equation. If the equation is true, the point lies on the plane. For instance, the point $(1,1,0)$ lies on the plane $2x-y+z=1$ because

$2(1)-1+0=1.$

The distance from a point to a plane is another idea that may appear in advanced geometry reasoning. While IB SL often focuses on identifying planes and working with their equations, the key logic is that the shortest distance is measured along a line perpendicular to the plane.

Relationships between lines and planes

Three-dimensional geometry is powerful because it helps us describe how objects relate to each other in space. A line may be parallel to a plane, lie in a plane, intersect a plane at one point, or be perpendicular to the plane.

A line is parallel to a plane if its direction vector is perpendicular to the plane’s normal vector. In dot product language, if $\mathbf{d}$ is a line direction vector and $\mathbf{n}$ is a plane normal vector, then

$\mathbf{d}\cdot\mathbf{n}=0$

means the line is parallel to the plane.

A line is perpendicular to a plane if its direction vector is parallel to the plane’s normal vector.

Two planes are parallel if their normal vectors are parallel. They are perpendicular if their normal vectors are perpendicular. That means the dot product of the normals is zero:

$\mathbf{n}_1\cdot\mathbf{n}_2=0.$

These tests are especially useful in exam questions because they turn geometric ideas into algebraic checks.

Example: consider the plane $x+2y-z=4$ and the plane $2x+4y-2z=1$.

Their normal vectors are

$\begin{pmatrix}1\\2\\-1\end{pmatrix}$

\quad\text{and}\quad

$\begin{pmatrix}2\\4\\-2\end{pmatrix}.$

The second is a scalar multiple of the first, so the planes are parallel.

Why this topic matters in Geometry and Trigonometry

Three-dimensional coordinate geometry connects directly to other parts of Geometry and Trigonometry. Distance formulas come from Pythagoras’ theorem. Angles between vectors rely on trigonometric ideas through the dot product formula. Lines and planes are studied using algebra, but the reasoning is still geometric.

This topic also supports later work with circular measure, trigonometric functions, and modeling, because space problems often require the same skill: represent the situation clearly, choose a formula, and interpret the result.

In IB Mathematics Analysis and Approaches SL, students, you are expected not just to memorize formulas but to use them in context. That means:

identifying the correct objects in space,
selecting the right equation or vector method,
checking whether answers make sense,
explaining your reasoning clearly.

A strong response in an exam often includes both calculation and interpretation. For example, after finding that two lines are parallel, you should state what that means geometrically, not just write the vector result.

Conclusion

Three-dimensional coordinate geometry extends the ideas of coordinate geometry into real space. It lets us describe points with $(x,y,z)$, calculate distances, find midpoints, write equations of lines and planes, and study how geometric objects relate. students, these tools are central to Geometry and Trigonometry because they combine algebra, geometry, and trigonometry in one framework.

If you can move confidently between words, diagrams, vectors, and equations, you have the main skills needed for this topic. Keep practicing with coordinates, direction vectors, and plane equations, and always connect your algebra back to the geometry it represents 📐.

Study Notes

A point in three dimensions is written as $(x,y,z)$.
The distance between $A(x_1,y_1,z_1)$ and $B(x_2,y_2,z_2)$ is

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

The midpoint of a segment from $A(x_1,y_1,z_1)$ to $B(x_2,y_2,z_2)$ is

$ \left(\frac{x_1+x_2}{2},\frac{y_1+y_2}{2},\frac{z_1+z_2}{2}\right).$

A vector from $A$ to $B$ is

$ \overrightarrow{AB}=\begin{pmatrix}x_2-x_1\y_2-y_1\z_2-z_1\end{pmatrix}.$

A line can be written as

$ \mathbf{r}=\mathbf{a}+t\mathbf{d}.$

A plane often has equation

$ ax+by+cz=d.$

The vector $\begin{pmatrix}a\b\c\end{pmatrix}$ is normal to the plane.
If $\mathbf{d}\cdot\mathbf{n}=0$, then the line direction vector $\mathbf{d}$ is parallel to the plane.
If two plane normals are parallel, the planes are parallel.
If two plane normals have dot product $0$, the planes are perpendicular.
This topic links algebraic methods with geometric reasoning in three dimensions.