How to Use Vectors in IGCSE Maths: Magnitude, Direction and Operations

Learning how to work with vectors in IGCSE Maths involves reading components correctly, combining vectors accurately, and using Pythagoras to calculate magnitudes. In IGCSE Maths, a vector is a quantity with both magnitude and direction, used to describe displacement, velocity and forces, as well as movement and transformations in geometry. This page acts as a central reference for the methods and notation expected in exam questions, supported by worked examples and structured practice.

Prior Knowledge This page requires confidence with Pythagoras' Theorem.
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Visualising a Vector

The vector \(\begin{pmatrix} 4 \\ 3 \end{pmatrix}\)

x y 0 1 2 3 4 5 6 0 1 2 3 4 O 4 units 3 units (4, 3)

The vector \( \begin{pmatrix} 4 \\ 3 \end{pmatrix} \) moves 4 units right and 3 units up. Only its length and direction matter, not its starting point.

Adding Vectors Geometrically

Head-to-tail addition: \(\mathbf{a} + \mathbf{b}\)

a (3, 1) b (1, 3) a + b

To add vectors, place \(\mathbf{a}\) and \(\mathbf{b}\) head-to-tail. The vector from the start of \(\mathbf{a}\) to the end of \(\mathbf{b}\) is the resultant \(\mathbf{a} + \mathbf{b}\).

What Is a Vector?

A vector is a "movement instruction" that tells you how far to move and which way to move. In 2D we usually write vectors as a column vector \( \begin{pmatrix} x \\ y \end{pmatrix} \) (you may also see the shorthand \((x, y)\)). In 3D we use \( \begin{pmatrix} x \\ y \\ z \end{pmatrix} \).

Step-by-Step: Reading a Vector

  1. Horizontal component: \(x\) tells you how many units left/right.
  2. Vertical component: \(y\) tells you how many units up/down.
  3. Direction: positive \(x\) = right, negative \(x\) = left; positive \(y\) = up, negative \(y\) = down.
  4. Magnitude: for \( \mathbf{v} = \begin{pmatrix} x \\ y \end{pmatrix}, \) use Pythagoras: \[ |\mathbf{v}| = \sqrt{x^2 + y^2}. \]

Core Ideas

Magnitude + direction
A vector is fully described by its magnitude (length) and direction. The starting point can move without changing the vector. That's why equal vectors can sit in different places on a diagram.
Component & column form
In 2D IGCSE questions, vectors are usually written as column vectors \( \begin{pmatrix} x \\ y \end{pmatrix} \). The top number is the x-component, the bottom is the y-component.
Add components
To add or subtract vectors, combine the components: \[ \begin{pmatrix} a \\ b \end{pmatrix} + \begin{pmatrix} c \\ d \end{pmatrix} = \begin{pmatrix} a+c \\ b+d \end{pmatrix}. \] Geometrically this is the head-to-tail rule.
Scalars & parallel vectors
A scalar \(k\) multiplies a vector: \( k\begin{pmatrix} x \\ y \end{pmatrix} = \begin{pmatrix} kx \\ ky \end{pmatrix}. \) If \(\mathbf{b} = k\mathbf{a}\) for some non-zero \(k\), then \(\mathbf{a}\) and \(\mathbf{b}\) are parallel.

Worked Examples

💡 Example 1: Magnitude of a Vector

Question: Find the magnitude of \( \mathbf{v} = \begin{pmatrix} -6 \\ 8 \end{pmatrix}. \)

\[ \begin{aligned} |\mathbf{v}| &= \sqrt{(-6)^2 + 8^2} \\ &= \sqrt{36 + 64} \\ &= \sqrt{100} \\ &= 10. \end{aligned} \]
What's happening?
  • Square each component separately (a negative squared is positive).
  • Add the squared values.
  • Take the square root to get the length.

Answer: \( |\mathbf{v}| = 10 \).

💡 Example 2: Adding Vectors

Question: Find the resultant of \( \mathbf{a} = \begin{pmatrix} 5 \\ -2 \end{pmatrix} \) and \( \mathbf{b} = \begin{pmatrix} -1 \\ 7 \end{pmatrix}. \)

\[ \begin{aligned} \mathbf{a} + \mathbf{b} &= \begin{pmatrix} 5 \\ -2 \end{pmatrix} + \begin{pmatrix} -1 \\ 7 \end{pmatrix} \\ &= \begin{pmatrix} 5 + (-1) \\ -2 + 7 \end{pmatrix} \\ &= \begin{pmatrix} 4 \\ 5 \end{pmatrix}. \end{aligned} \]
What's happening?
  • Add x-components together.
  • Add y-components together.
  • The resultant vector is \( \begin{pmatrix} 4 \\ 5 \end{pmatrix}. \)

Geometrically, the resultant is the diagonal when you place the vectors head-to-tail (as in the addition diagram above).

💡 Example 3: Scalar Multiplication

Question: Find \( -2\mathbf{v} \) if \( \mathbf{v} = \begin{pmatrix} 3 \\ -1 \end{pmatrix}. \)

\[ \begin{aligned} -2\mathbf{v} &= -2 \begin{pmatrix} 3 \\ -1 \end{pmatrix} \\ &= \begin{pmatrix} -6 \\ 2 \end{pmatrix}. \end{aligned} \]
What's happening?
  • Multiply each component by \(-2\).
  • The vector doubles in length.
  • The negative sign reverses its direction.

Multiplying by a positive scalar stretches or shrinks the vector; a negative scalar also reverses its direction.

💡 Example 4: Unit Vector in the Same Direction

Question: Find the unit vector in the direction of \( \mathbf{v} = \begin{pmatrix} 5 \\ 12 \end{pmatrix}. \)

\[ \begin{aligned} |\mathbf{v}| &= \sqrt{5^2 + 12^2} \\ &= \sqrt{25 + 144} \\ &= \sqrt{169} \\ &= 13. \\[6pt] \hat{\mathbf{v}} &= \frac{\mathbf{v}}{|\mathbf{v}|} \\ &= \begin{pmatrix} \dfrac{5}{13} \\[4pt] \dfrac{12}{13} \end{pmatrix}. \end{aligned} \]
What's happening?
  • Use the Pythagorean triple \(5,12,13\) to find the magnitude quickly.
  • Divide each component by 13.
  • The new vector has length 1 but keeps the same direction.

The result \( \begin{pmatrix} 5/13 \\ 12/13 \end{pmatrix} \) has magnitude 1 and points the same way as \( \mathbf{v} \): that is what "unit vector" means.

IGCSE Vector Notation You'll See in Questions

  • Bold letters: \(\mathbf{a}, \mathbf{b}, \mathbf{v}\) are vectors. Scalars (ordinary numbers) are written in normal type like \(2, -3, k\).
  • Column vectors: exam answers are usually expected in the form \( \begin{pmatrix} x \\ y \end{pmatrix} \), even if the question starts with \((x, y)\).
  • Position vectors: the position vector of a point \(P\) is the vector from the origin \(O\) to \(P\), written \( \overrightarrow{OP} \).
  • Vector \( \overrightarrow{AB} \): means "from \(A\) to \(B\)". If \(A\) has position vector \( \mathbf{a} \) and \(B\) has position vector \( \mathbf{b} \), then \( \overrightarrow{AB} = \mathbf{b} - \mathbf{a}. \)
  • Parallel vectors: if \( \mathbf{b} = k\mathbf{a} \) for some non-zero scalar \(k\), then \(\mathbf{a}\) and \(\mathbf{b}\) are parallel and point along the same line.
  • Algebra with \(\mathbf{a}\) and \(\mathbf{b}\): you'll often see questions like "Let \( \mathbf{a} = \begin{pmatrix} 2 \\ 3 \end{pmatrix}, \mathbf{b} = \begin{pmatrix} -1 \\ 4 \end{pmatrix} \). Find \(2\mathbf{a} + \mathbf{b}\)." Just work with the components.

🔑 Key Points

  • A vector has both magnitude and direction.
  • Component form \( \begin{pmatrix} x \\ y \end{pmatrix} \) tells you horizontal and vertical movement.
  • Add and subtract vectors by combining components.
  • Use Pythagoras to find magnitude: \( |\mathbf{v}| = \sqrt{x^2 + y^2}. \)
  • Unit vectors have length 1 and show pure direction; parallel vectors are scalar multiples of each other.

⚠️ Common Pitfalls

  • Forgetting to square each component when finding magnitude.
  • Mixing up the order of components (accidentally writing \( (y, x) \)).
  • Adding magnitudes instead of adding components.
  • Forgetting that a negative scalar reverses direction.
  • Dropping surds too early when an exact answer is required.
  • Thinking \( \overrightarrow{AB} = \overrightarrow{BA} \): in fact \( \overrightarrow{BA} = -\overrightarrow{AB}. \)

Comfortable with basic vectors? Next, explore how vectors are used in three dimensions and how Pythagoras and trigonometry extend into 3D problems.

Next: 3D Trigonometry (IGCSE)

Vectors: Practice Room

Practise core IGCSE vector arithmetic skills: reading components, adding and subtracting vectors, scalar multiples, magnitudes, position vectors and parallel vectors. These questions match the Vectors (IGCSE) notes above. Column-vector answers have two boxes: the top box is the x-component and the bottom box is the y-component. Magnitudes go in a single box, as a whole number where the answer is a whole number and as a surd such as sqrt(29) for \( \sqrt{29} \) where it is not. Fractions go in as p/q.

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Input tips: column vectors go in the two boxes (x on top, y below), so \( \begin{pmatrix} 3 \\ 4 \end{pmatrix} \) is 3 then 4, and a unit vector is 3/5 then 4/5. Magnitudes go in one box as sqrt(29) (or \sqrt{29}, or 2sqrt(2) in simplest form). Spacing is ignored.

Beginners guide to Vectors

This YouTube video provides a comprehensive introduction to vectors. It covers the basic concepts of vectors, their representation, operations, and properties. The video explains vector addition, subtraction, and scalar multiplication, as well as the concept of the dot product. By watching this video, you will gain a solid foundation in vectors and their fundamental operations.

Part 2: Looking at more complex problems

In this video we kick it up a gear looking at more complex vector problems involving ratio and mid points building on the concepts that were introduced in part 1.