Historical Context & Motivation
Imagine you are giving a friend directions to a coffee shop. You wouldn't just say "walk 500 metres" — you'd also need to say in which direction. That distinction — needing both a size and a direction — is exactly the problem that vectors were invented to solve. For centuries, scientists and mathematicians struggled to combine forces, velocities, and displacements that pointed in different directions. Ordinary numbers (which we call scalars) only capture size, so a new kind of mathematical object was needed.
The central question that drove all this work was deceptively simple: how do we add, subtract, and measure quantities that point in different directions? In this lesson, you will learn the modern answer — representing vectors by their components, calculating their magnitude, and performing basic operations.
Core Principles & Definitions
Before diving into calculations, let's establish the foundational ideas. A vector is a mathematical object that carries two pieces of information: how much (its magnitude) and which way (its direction). Examples include velocity, force, and displacement. By contrast, a scalar only has magnitude — think of temperature, mass, or speed (speed is the magnitude of velocity).
Components
Magnitude
Direction
Equality of Vectors
Basic Operations
Visualising Vectors on the Coordinate Plane
The best way to understand vectors is to see them drawn on a coordinate plane. Every 2D vector can be represented as an arrow starting at the origin and ending at the point defined by its components. The diagram below shows the vector a = (4, 3), illustrating how the x-component and y-component form a right triangle with the vector itself as the hypotenuse.
Notice how the vector, its x-component, and its y-component form a right triangle. This is the key geometric insight: because the components are perpendicular, we can use the Pythagorean theorem to find the vector's magnitude. The angle θ (shown in green) represents the direction of the vector, measured anti-clockwise from the positive x-axis. You can always recover this angle using trigonometry once you know the components.
Mathematical Framework
Now let's formalise the ideas from the diagram into equations. In the IB course, vectors are typically written as column vectors or in component form. Below are the essential formulas you need for SL 3.5.
Vector Addition & Subtraction — A Closer Look
Understanding vector addition visually is just as important as doing it algebraically. There are two equivalent geometric methods: the tip-to-tail method and the parallelogram method. The diagram below shows both, applied to the same pair of vectors.
Vector subtraction works the same way, but you first reverse the direction of the vector being subtracted. That is, a − b = a + (−b). In component form: (aₓ − bₓ , ay − by). Geometrically, −b is the vector b flipped 180°, so subtracting is just adding the reversed vector.
| Operation | Algebraic Rule | Geometric Picture |
|---|---|---|
| Addition | (aₓ + bₓ , a_y + b_y) | Tip-to-tail or parallelogram diagonal |
| Subtraction | (aₓ − bₓ , a_y − b_y) | Add the reversed vector −b to a |
| Scalar × | (k × vₓ , k × v_y) | Stretch / shrink; negative k flips direction |
Worked Example
Let's walk through a complete problem that combines several skills: finding a resultant vector, computing its magnitude, and interpreting the result.
Vectors vs. Scalars — Strengths & Limitations
It's worth pausing to see why vectors are so useful — and where they have their limits at the SL level. The table below compares vector quantities with scalar quantities across several categories.
| Feature | Scalar | Vector |
|---|---|---|
| Information carried | Magnitude only (e.g. 5 km) | Magnitude and direction (e.g. 5 km east) |
| Addition | Ordinary arithmetic (3 + 4 = 7) | Component-wise; result depends on directions |
| Examples | Temperature, mass, speed, energy | Velocity, force, displacement, acceleration |
| Can be negative? | Yes (e.g. −5 °C) | Components can be negative (direction, not size) |
| Representation | A single number with units | An arrow, a pair of components, or column notation |
Connection to Advanced Theory
The 2D vector toolkit you've learned in this lesson is the foundation for more advanced topics. At the HL level, vectors expand into three dimensions and you encounter operations like the scalar (dot) product and the vector (cross) product. Even within SL, vectors reappear when you study kinematics (velocity and acceleration as vectors) and vector equations of lines.
| Concept | SL 3.5 (This Lesson) | HL / Further Study |
|---|---|---|
| Dimensions | 2D: (x, y) | 3D: (x, y, z) and beyond |
| Operations | Addition, subtraction, scalar multiplication, magnitude | Dot product, cross product, unit vectors, projections |
| Applications | Displacement, force diagrams, navigation | 3D geometry, physics simulations, computer graphics |
| Notation | Column vectors, component pairs | î, ĵ, k̂ unit vector notation; matrix representations |
Mastering the component approach and the magnitude formula now will make the transition to these more advanced ideas smooth. Every new operation — whether it's a dot product or a vector equation of a line — builds directly on the skills of reading, writing, and manipulating components that you practised in this lesson.
Practice Problems
Lesson Summary
A 2D vector carries both magnitude and direction, written in component form as (vₓ, vy). The magnitude is found using the Pythagorean theorem: |v| = √(vₓ² + vy²). Vector addition and subtraction are performed component-by-component, while scalar multiplication scales every component by the same factor.
Geometrically, addition uses the tip-to-tail or parallelogram method. These foundational skills — reading components, computing magnitude, and combining vectors — form the basis for every further vector topic in the IB, from dot products to vector equations of lines. Remember: scalars answer 'how much?'; vectors answer 'how much and which way?'