What you'll learn
This topic covers vectors — quantities with both size and direction — and the operations you can do with them. In this guide you will learn vector notation, how to add and subtract vectors, how to multiply by a scalar, how to find the magnitude, and how to use vectors in geometric proofs. These are important higher-tier skills linking algebra and geometry.
Key terms and definitions
Vector — a quantity with magnitude and direction.
Column vector — a vector written as a top (x) and bottom (y) number.
Scalar — an ordinary number that scales a vector.
Magnitude — the length of a vector.
Resultant — the single vector equal to two or more combined.
Core concepts
Vector notation
A vector has size and direction, written as a, $\vec{AB}$, or a column vector with the horizontal part on top and the vertical part below. $\vec{AB}$ means "from A to B"; reversing direction gives $\vec{BA} = -\vec{AB}$.
Adding and subtracting vectors
To add column vectors, add the top numbers and the bottom numbers separately; to subtract, subtract them. Geometrically, addition follows "nose to tail" to give the resultant, and $\vec{AB} = \vec{AC} + \vec{CB}$ lets you travel via another point.
Scalar multiplication
Multiplying by a scalar multiplies both parts of the vector, changing its length but keeping (or reversing) its direction. For example, 3a is three times as long as a in the same direction; −2a is twice as long in the opposite direction. Parallel vectors are scalar multiples of each other.
Magnitude
The magnitude (length) of a column vector is found with Pythagoras: for $\begin{pmatrix}x\y\end{pmatrix}$, magnitude = √(x² + y²). This gives the actual distance the vector represents.
Vectors in geometry proofs
Vectors prove geometric facts: writing journeys in terms of base vectors lets you show lines are parallel (one is a scalar multiple of the other) or that points are collinear (on the same straight line). Express each path through known vectors and simplify.
Worked examples
Example 1: Adding
Add $\begin{pmatrix}3\1\end{pmatrix}$ and $\begin{pmatrix}2\-4\end{pmatrix}$.
$\begin{pmatrix}5\-3\end{pmatrix}$.
Example 2: Scalar multiple
Find 3 times $\begin{pmatrix}2\-1\end{pmatrix}$.
$\begin{pmatrix}6\-3\end{pmatrix}$.
Example 3: Magnitude
Find the magnitude of $\begin{pmatrix}3\4\end{pmatrix}$.
√(3² + 4²) = √25 = 5.
Common mistakes and how to avoid them
Adding the wrong parts. Add top to top and bottom to bottom.
Sign error reversing direction. $\vec{BA} = -\vec{AB}$.
Only scaling one part. A scalar multiplies both components.
Forgetting Pythagoras for magnitude. Use √(x² + y²).
Not simplifying in proofs. Combine vectors fully to reveal scalar multiples.
Exam technique for Vectors
Add and subtract components separately.
Multiply both parts by a scalar.
Use $\vec{AB} = \vec{AC} + \vec{CB}$ to route through points.
Find magnitude with Pythagoras.
Show one vector is a multiple of another to prove parallel or collinear.
Quick revision summary
A vector has magnitude and direction, written as a, $\vec{AB}$, or a column vector (horizontal on top, vertical below), with $\vec{BA} = -\vec{AB}$. Add or subtract column vectors component by component; geometrically addition is "nose to tail" giving the resultant, and $\vec{AB} = \vec{AC} + \vec{CB}$ routes via another point. Scalar multiplication multiplies both parts, changing length and possibly reversing direction — and parallel vectors are scalar multiples of each other. The magnitude is found with Pythagoras, √(x² + y²). In proofs, express journeys through base vectors to show lines are parallel (one a multiple of another) or points collinear. The common errors are adding the wrong parts, sign slips when reversing, scaling only one component, and forgetting Pythagoras for magnitude. Work component by component, scale both parts, route through points, use Pythagoras for length, and spot scalar multiples in proofs.