Historical Context & Motivation
For centuries, mathematicians and scientists worked with quantities that could only be described by their size β think of temperature or mass. However, many physical phenomena, such as forces acting on a bridge or the velocity of a spacecraft, require both a size and a direction. The concept of a vector was developed to capture exactly this dual nature. While 2D vectors are powerful for describing motion in a flat plane, the world we inhabit is fundamentally three-dimensional, and extending vectors into 3D space unlocked an entirely new level of mathematical precision.
The central question for this lesson is: how do we precisely describe a quantity that has both magnitude and direction in the three-dimensional space we live in? To answer that, we need to understand 3D vector components, how to compute magnitude, and how to define unit vectors that capture direction alone.
Core Principles & Definitions
Before diving into calculations, you need a solid grasp of the foundational ideas that underpin 3D vectors. Each principle below builds on concepts you already know from 2D vectors but extends them by adding a third axis β the z-axis β perpendicular to both the x-axis and the y-axis.
3D Coordinate System
Vector Components
Magnitude (Length)
Unit Vectors
Standard Basis Vectors
Visualising Vectors in 3D Space
One of the trickiest parts of working with 3D vectors is visualising them on a flat page or screen. The diagram below shows a right-hand coordinate system with the three axes and a vector v = (3, 4, 5) drawn from the origin. Notice how the vector's components correspond to projections onto each axis β the dashed lines show the "shadow" of the vector along each direction.
When you look at this diagram, notice that the x-axis and y-axis form the "floor" of the coordinate system, while the z-axis points upward. The right-hand rule is a convention that determines the positive direction of the z-axis: if you curl the fingers of your right hand from the positive x-axis toward the positive y-axis, your thumb points along the positive z-axis. This convention ensures consistency across all IB problems and scientific applications.
Mathematical Framework
Now let's formalise the key formulas you need for 3D vectors. These equations are the tools that turn the visual intuition from the previous section into precise, calculable results.
Position Vectors & Components
If a vector connects two points A = (aβ, aβ, aβ) and B = (bβ, bβ, bβ), then the vector from A to B is found by subtracting coordinates component-wise.
Magnitude of a 3D Vector
Unit Vectors
Components, Basis Vectors & Scalar Multiplication
Understanding the anatomy of a 3D vector goes beyond simply listing three numbers. Each component represents a scalar multiple of the corresponding standard basis vector. For instance, the vector (3, β2, 7) is really 3π’ + (β2)π£ + 7π€. This decomposition makes it easy to add, subtract, or scale vectors β you simply perform the operation on each component independently.
Scalar Multiplication in 3D
When you multiply a 3D vector by a scalar k, you multiply each component by k. If k > 0, the vector stretches (or shrinks) but keeps its direction; if k < 0, the direction reverses. Formally, k Β· v = (k Β· vβ, k Β· vβ, k Β· vβ). This is exactly how the unit vector formula works β dividing by |v| is the same as multiplying by the scalar 1/|v|.
| Operation | Formula | Result Type |
|---|---|---|
| Addition | u + v = (uβ+vβ, uβ+vβ, uβ+vβ) | Vector |
| Subtraction | u β v = (uββvβ, uββvβ, uββvβ) | Vector |
| Scalar multiplication | kv = (kvβ, kvβ, kvβ) | Vector |
| Magnitude | |v| = β(vβΒ² + vβΒ² + vβΒ²) | Scalar (β₯ 0) |
| Unit vector | vΜ = v / |v| | Vector (|vΜ| = 1) |
Worked Example
Let's work through a complete example that ties together components, magnitude, and the unit vector.
2D Vectors vs 3D Vectors β Comparison
If you're already comfortable with 2D vectors, moving to 3D is largely a matter of adding one more component. Most of the formulas look almost identical β they just have an extra term. The table below highlights the key similarities and differences.
| Feature | 2D Vectors | 3D Vectors |
|---|---|---|
| Components | (vβ, vβ) | (vβ, vβ, vβ) |
| Basis vectors | π’, π£ | π’, π£, π€ |
| Magnitude | β(vβΒ² + vβΒ²) | β(vβΒ² + vβΒ² + vβΒ²) |
| Direction | One angle ΞΈ suffices | Two angles (or direction cosines) needed |
| Visualisation | Easy on paper (flat plane) | Requires perspective/projection drawings |
| Cross product | Not truly defined (only a scalar) | Produces a perpendicular 3D vector |
Connection to Advanced Topics
The concepts of components, magnitude, and unit vectors are the foundation for everything else you'll study in AHL vector work. Once you master these basics, the IB syllabus builds directly on them with the scalar (dot) product, the vector (cross) product, and applications to lines and planes in 3D. The table below previews how today's content connects to these more advanced ideas.
| This Lesson | Where It Leads |
|---|---|
| Components (vβ, vβ, vβ) | Used to compute dot products (v Β· w = vβwβ + vβwβ + vβwβ) and cross products |
| Magnitude |v| | Appears in the angle formula: cos ΞΈ = (v Β· w) / (|v| Γ |w|) |
| Unit vectors vΜ | Direction vectors for lines in 3D; normal vectors for planes |
| Vector between two points | Parametric equations of lines: r = a + t Β· d |
| Standard basis π’, π£, π€ | Determinant form of the cross product uses these as a row |
Beyond the IB, 3D vectors are essential in physics (describing forces, fields, and torques), in computer graphics (placing objects in virtual worlds), and in engineering (stress analysis, robotics). If you continue to university-level mathematics, vectors generalise into n-dimensional vector spaces where the same ideas of magnitude and unit vectors apply β just with more components.
Practice Problems
Lesson Summary
A 3D vector is represented by three components (vβ, vβ, vβ), each measuring displacement along one of the three perpendicular axes in a right-hand coordinate system. Every 3D vector can be written as a linear combination of the standard basis vectors π’, π£, and π€. The magnitude is computed as |v| = β(vβΒ² + vβΒ² + vβΒ²), extending the Pythagorean theorem into three dimensions.
A unit vector vΜ = v / |v| preserves the direction of v while scaling its length to exactly 1. To find a vector between two points, subtract coordinates: ABβ = B β A. These foundational skills β writing components, computing magnitudes, and constructing unit vectors β are the building blocks for every advanced vector topic in the IB HL syllabus, including dot products, cross products, and equations of lines and planes in three-dimensional space.