Three-Dimensional Geometry

Introduction

students, have you ever looked at a bridge, a skyscraper, or a shipping container and wondered how engineers know its size, angle, and position in space? 🌉📦 That is the power of three-dimensional geometry. In this lesson, you will learn how to describe and measure objects in space using points, lines, planes, vectors, distances, and angles. These ideas are central to $\text{IB Mathematics: Applications and Interpretation HL}$ because they connect geometry with real-world design, navigation, and modelling.

By the end of this lesson, you should be able to:

explain the key language of three-dimensional geometry,
work with coordinates and vectors in space,
calculate distances, angles, and equations for lines and planes,
connect spatial reasoning to practical problems,
and see how this topic fits into $\text{Geometry and Trigonometry}$.

Three-dimensional geometry is not just about drawing shapes. It is about reasoning in space, where position and direction matter. That is why it is useful in architecture, robotics, satellite positioning, and game design 🎮.

Coordinates, Points, and Vectors in Space

In three dimensions, a point is written as $\left(x,y,z\right)$. The three coordinates tell you how far the point is from the origin along the $x$-, $y$-, and $z$-axes. The origin is $\left(0,0,0\right)$.

A vector in space is often written as $\begin{pmatrix}a\b\c\end{pmatrix}$ or as $a\mathbf{i}+b\mathbf{j}+c\mathbf{k}$. Vectors show direction and magnitude. For example, the vector from $A\left(1,2,3\right)$ to $B\left(4,6,8\right)$ is

$$\overrightarrow{AB}=\begin{pmatrix}4-1\\6-2\\8-3\end{pmatrix}=\begin{pmatrix}3\\4\\5\end{pmatrix}$$

This means you move $3$ units in the $x$-direction, $4$ in the $y$-direction, and $5$ in the $z$-direction.

A key idea is that a vector can be moved anywhere in space without changing it, as long as its length and direction stay the same. This is different from a point, which has a fixed location.

Example: if a drone flies from $P\left(2,1,0\right)$ to $Q\left(5,5,4\right)$, then its displacement vector is

$$\overrightarrow{PQ}=\begin{pmatrix}3\\4\\4\end{pmatrix}$$

This kind of reasoning helps in navigation systems, where movement is described in 3D rather than on a flat map.

Distances and Midpoints in Three Dimensions

One of the most important tasks in three-dimensional geometry is measuring distance. The distance between two points $A\left(x_1,y_1,z_1\right)$ and $B\left(x_2,y_2,z_2\right)$ is found using

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

This is the 3D version of the distance formula from coordinate geometry. It comes from the same Pythagorean idea, but applied twice.

Example: find the distance between $A\left(1,2,3\right)$ and $B\left(5,5,7\right)$. Substitute into the formula:

$$AB=\sqrt{\left(5-1\right)^2+\left(5-2\right)^2+\left(7-3\right)^2}$$

$$AB=\sqrt{16+9+16}=\sqrt{41}$$

So the points are separated by $\sqrt{41}$ units.

The midpoint of a segment is also important. If $A\left(x_1,y_1,z_1\right)$ and $B\left(x_2,y_2,z_2\right)$, 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 in locating the center of a beam, a cable, or a route between two points.

For example, the midpoint between $\left(2,4,6\right)$ and $\left(8,0,10\right)$ is

$$\left(\frac{2+8}{2},\frac{4+0}{2},\frac{6+10}{2}\right)=\left(5,2,8\right)$$

Lines and Planes in Space

A line in 3D can be described using a point and a direction vector. If a line passes through $\mathbf{r}_0$ and has direction vector $\mathbf{d}$, then its vector equation is

$$\mathbf{r}=\mathbf{r}_0+t\mathbf{d}$$

where $t$ is a real number. This means every point on the line can be reached by starting at one point and moving some multiple of the direction vector.

Example: a line through $\left(1,2,3\right)$ with direction vector $\begin{pmatrix}2\\-1\\4\end{pmatrix}$ has vector equation

$$\mathbf{r}=\begin{pmatrix}1\\2\\3\end{pmatrix}+t\begin{pmatrix}2\\-1\\4\end{pmatrix}$$

Planes are flat surfaces that extend forever in all directions. A plane can be written in Cartesian form as

$$ax+by+cz=d$$

where $a$, $b$, and $c$ are constants. The vector $\begin{pmatrix}a\b\c\end{pmatrix}$ is normal to the plane, which means it is perpendicular to the plane.

If a plane has equation

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

then a normal vector is

$$\begin{pmatrix}2\\-3\\1\end{pmatrix}$$

A line and a plane may intersect, be parallel, or a line may lie entirely in a plane. This makes spatial reasoning richer than 2D geometry because there are more possible relationships.

Example of a practical interpretation: in a building model, a wall can be treated as a plane, while a support rod can be treated as a line. Knowing whether the rod hits the wall at a right angle helps in design and stability.

Angles Between Lines and Planes

Angle calculations are a major skill in this topic. The angle between two vectors $\mathbf{a}$ and $\mathbf{b}$ is found using the dot product formula:

$$\mathbf{a}\cdot\mathbf{b}=\left|\mathbf{a}\right|\left|\mathbf{b}\right|\cos\theta$$

where $\theta$ is the angle between them.

If $\mathbf{a}=\begin{pmatrix}1\\2\\2\end{pmatrix}$ and $\mathbf{b}=\begin{pmatrix}2\\0\\1\end{pmatrix}$, then

$$\mathbf{a}\cdot\mathbf{b}=1\cdot2+2\cdot0+2\cdot1=4$$

Also,

$$\left|\mathbf{a}\right|=\sqrt{1^2+2^2+2^2}=3$$

and

$$\left|\mathbf{b}\right|=\sqrt{2^2+0^2+1^2}=\sqrt{5}$$

$$\cos\theta=\frac{4}{3\sqrt{5}}$$

This angle idea is used for finding the tilt of a roof, the direction of a satellite path, or the meeting angle of two roads in space.

The angle between a line and a plane is related to the angle between the line and the plane’s normal vector. If a line is perpendicular to the normal vector, then the line is parallel to the plane. If the line has the same direction as the normal vector, then it is perpendicular to the plane.

Why Three-Dimensional Geometry Matters

Three-dimensional geometry links many ideas from $\text{Geometry and Trigonometry}$: distance, angle, vectors, and measurement. It also supports modelling, which is a major part of $\text{Applications and Interpretation HL}$. In real life, dimensions are not usually flat. A delivery drone flies through space, not along a page. A staircase rises as well as runs. A satellite orbits in a 3D system.

A good way to think about this topic is: coordinates locate, vectors direct, distance measures, and equations describe. Together, these tools help you reason accurately about objects in space.

Consider a simple example. Suppose a package travels from $A\left(0,0,0\right)$ to $B\left(6,2,3\right)$. Its displacement vector is

$$\begin{pmatrix}6\\2\\3\end{pmatrix}$$

The length of this movement is

$$\sqrt{6^2+2^2+3^2}=\sqrt{49}=7$$

So the straight-line path is $7$ units long. This is exactly the kind of result that helps in robotics, transport planning, and spatial design.

Conclusion

Three-dimensional geometry extends familiar ideas from 2D into space. students, you now have the key language to describe points, vectors, lines, planes, distances, and angles in $3$D. These ideas are essential for mathematical modelling and for understanding how objects fit and move in the real world. In $\text{IB Mathematics: Applications and Interpretation HL}$, this topic strengthens your ability to reason spatially and solve applied problems with precision 📐.

Study Notes

A point in space is written as $\left(x,y,z\right)$.
A vector shows direction and magnitude, for example $\begin{pmatrix}a\b\c\end{pmatrix}$.
The vector from $A\left(x_1,y_1,z_1\right)$ to $B\left(x_2,y_2,z_2\right)$ is $\begin{pmatrix}x_2-x_1\y_2-y_1\z_2-z_1\end{pmatrix}$.
The distance between two points is $\sqrt{\left(x_2-x_1\right)^2+\left(y_2-y_1\right)^2+\left(z_2-z_1\right)^2}$.
The midpoint of a segment is $\left(\frac{x_1+x_2}{2},\frac{y_1+y_2}{2},\frac{z_1+z_2}{2}\right)$.
A line can be written as $\mathbf{r}=\mathbf{r}_0+t\mathbf{d}$.
A plane can be written as $ax+by+cz=d$.
The vector $\begin{pmatrix}a\b\c\end{pmatrix}$ is normal to the plane $ax+by+cz=d$.
The dot product helps find angles using $\mathbf{a}\cdot\mathbf{b}=\left|\mathbf{a}\right|\left|\mathbf{b}\right|\cos\theta$.
Three-dimensional geometry is important for real-world measurement, navigation, engineering, and spatial reasoning.