Vectors

Further Mathematics 2330 · CCEA GCSE · 7 min read · Updated: 31 Aug 2026

Welcome to Vectors in Mechanics!

Welcome to one of the most useful and exciting tools in Mechanics: Vectors! If you have ever given someone directions like "walk 50 metres East" or pushed a supermarket trolley with a certain force in a specific direction, you have already used vectors in real life.

In this chapter, we will bridge the gap between simple numbers and real-world movement. Vectors allow us to describe not just how much something is moving or being pushed, but in which direction it is happening. Don't worry if this seems tricky at first; we will break everything down step-by-step with plenty of clear examples!

1. Scalars vs Vectors: What is the Difference?

In physics and mechanics, every quantity we measure falls into one of two categories: scalars or vectors.

Scalar Quantities: A scalar has magnitude (size) only. It does not have a direction attached to it.
• Examples: Time (\(12\text{ s}\)), Mass (\(5\text{ kg}\)), Distance (\(100\text{ m}\)), Speed (\(20\text{ m s}^{-1}\)), Temperature (\(18^\circ\text{C}\)).

Vector Quantities: A vector has both magnitude (size) and direction.
• Examples: Displacement (\(100\text{ m}\) North), Velocity (\(20\text{ m s}^{-1}\) at \(045^\circ\)), Acceleration (\(9.8\text{ m s}^{-2}\) downwards), Force (\(50\text{ N}\) to the right).

Did you know? Speed and Velocity are not the same thing in mechanics! Speed is a scalar (just how fast you are going), while velocity is a vector (how fast you are going and the direction you are travelling).

Key Takeaway: If a measurement needs a direction to make full sense in the real world, it is a vector.

2. Vector Notation: How We Write Vectors

In CCEA GCSE Further Mathematics, you will meet two main ways to write 2D vectors:

A. Column Vector Notation

A vector can be written as a vertical column:
\(\mathbf{a} = \begin{pmatrix} x \\ y \end{pmatrix}\)
• The top number, \(x\), represents the horizontal step (positive means right, negative means left).
• The bottom number, \(y\), represents the vertical step (positive means up, negative means down).

B. Unit Vector Notation (\(\mathbf{i}\) and \(\mathbf{j}\))

We define two special base unit vectors, each of length \(1\):
• \(\mathbf{i} = \begin{pmatrix} 1 \\ 0 \end{pmatrix}\) is a unit vector of length \(1\) unit in the positive \(x\)-direction (horizontal).
• \(\mathbf{j} = \begin{pmatrix} 0 \\ 1 \end{pmatrix}\) is a unit vector of length \(1\) unit in the positive \(y\)-direction (vertical).

Any vector \(\begin{pmatrix} x \\ y \end{pmatrix}\) can be written as: \(\mathbf{r} = x\mathbf{i} + y\mathbf{j}\).
Example: \(\begin{pmatrix} 4 \\ -3 \end{pmatrix} = 4\mathbf{i} - 3\mathbf{j}\).

Common Mistake to Avoid: When writing vectors by hand in an exam, write a little underline beneath letters (like \(\underline{a}\), \(\underline{i}\), \(\underline{j}\)) because you cannot write in bold font like a printed textbook!

Key Takeaway: Column vectors \(\begin{pmatrix} x \\ y \end{pmatrix}\) and component form \(x\mathbf{i} + y\mathbf{j}\) represent the exact same thing.

3. Basic Vector Arithmetic

Working with vectors is just like working with ordinary numbers, provided you keep the horizontal components (\(\mathbf{i}\)) and vertical components (\(\mathbf{j}\)) separate.

Adding and Subtracting Vectors

To add or subtract vectors, simply add or subtract the corresponding components.

If \(\mathbf{a} = 3\mathbf{i} + 5\mathbf{j}\) and \(\mathbf{b} = 2\mathbf{i} - 8\mathbf{j}\):
• \(\mathbf{a} + \mathbf{b} = (3 + 2)\mathbf{i} + (5 + (-8))\mathbf{j} = 5\mathbf{i} - 3\mathbf{j}\)
• \(\mathbf{a} - \mathbf{b} = (3 - 2)\mathbf{i} + (5 - (-8))\mathbf{j} = 1\mathbf{i} + 13\mathbf{j} = \mathbf{i} + 13\mathbf{j}\)

Multiplying by a Scalar

To multiply a vector by an ordinary number (a scalar), multiply each component by that number.
If \(\mathbf{a} = 4\mathbf{i} - 6\mathbf{j}\), then \(3\mathbf{a} = 3(4\mathbf{i} - 6\mathbf{j}) = 12\mathbf{i} - 18\mathbf{j}\).

Key Takeaway: Always treat horizontal (\(\mathbf{i}\)) and vertical (\(\mathbf{j}\)) directions as two independent tracks that never mix directly during addition and subtraction.

4. Magnitude and Direction of a Vector

Every vector has two core properties: magnitude (its length) and direction (the angle it makes).

Calculating Magnitude

Because the \(\mathbf{i}\) and \(\mathbf{j}\) components are at right angles (\(90^\circ\)) to each other, we use Pythagoras' Theorem to find the magnitude (length) of a vector.

The magnitude of a vector \(\mathbf{r} = x\mathbf{i} + y\mathbf{j}\) is written with vertical bars \(|\mathbf{r}|\):
\(|\mathbf{r}| = \sqrt{x^2 + y^2}\)

Worked Example: Find the magnitude of the velocity vector \(\mathbf{v} = 6\mathbf{i} - 8\mathbf{j}\text{ m s}^{-1}\).
Step 1: Square both components: \(6^2 = 36\) and \((-8)^2 = 64\).
Step 2: Add them together: \(36 + 64 = 100\).
Step 3: Take the square root: \(|\mathbf{v}| = \sqrt{100} = 10\text{ m s}^{-1}\).

Calculating Direction

To find the angle \(\theta\) a vector makes with a reference line (such as the positive horizontal \(x\)-axis or the vector \(\mathbf{i}\)), we use simple trigonometry:

\(\tan \theta = \left|\frac{\text{vertical component}}{\text{horizontal component}}\right| = \left|\frac{y}{x}\right|\)
\(\theta = \tan^{-1}\left|\frac{y}{x}\right|\)

Top Tip: Always sketch a small right-angled triangle showing the \(x\) and \(y\) arrows! This makes it immediately clear whether your angle is above or below the horizontal, or what bearing it represents.

Key Takeaway: Magnitude uses Pythagoras (\(\sqrt{x^2 + y^2}\)) and direction uses Trigonometry (\(\tan^{-1}\left|\frac{y}{x}\right|\)).

5. Position Vectors and Displacement

In mechanics, we often track the location of moving objects on a coordinate plane.

Position Vector: The position vector of a point \(A(x, y)\) is the vector from the fixed origin \(O(0, 0)\) to \(A\), written as \(\vec{OA}\) or \(\mathbf{r}_A\):
\(\vec{OA} = x\mathbf{i} + y\mathbf{j}\)

Displacement between Two Points: To find the vector displacement from point \(A\) to point \(B\), we use the vector triangle rule:
\(\vec{AB} = \vec{OB} - \vec{OA} = \mathbf{r}_B - \mathbf{r}_A\)

Worked Example: A drone moves from position \(A\) with position vector \(2\mathbf{i} + 3\mathbf{j}\text{ m}\) to position \(B\) with position vector \(7\mathbf{i} - 9\mathbf{j}\text{ m}\). Find the displacement vector \(\vec{AB}\) and the straight-line distance between them.
Step 1 (Displacement):
\(\vec{AB} = (7\mathbf{i} - 9\mathbf{j}) - (2\mathbf{i} + 3\mathbf{j})\)
\(\vec{AB} = (7 - 2)\mathbf{i} + (-9 - 3)\mathbf{j} = 5\mathbf{i} - 12\mathbf{j}\text{ m}\)
Step 2 (Distance): Distance is simply the magnitude of displacement:
\(|\vec{AB}| = \sqrt{5^2 + (-12)^2} = \sqrt{25 + 144} = \sqrt{169} = 13\text{ m}\).

Key Takeaway: The vector from \(A\) to \(B\) is always \(\text{End} - \text{Start}\) (\(\vec{OB} - \vec{OA}\)).

6. Vectors in Mechanics: Velocity, Acceleration, and Forces

Now let's apply vectors to mechanics problems involving movement and forces!

A. Constant Velocity and Motion

When an object moves with constant velocity \(\mathbf{v}\), its displacement after time \(t\) seconds is given by:
\(\mathbf{s} = \mathbf{v}t\)

If the object starts from an initial position vector \(\mathbf{r}_0\), its position vector \(\mathbf{r}\) at time \(t\) is:
\(\mathbf{r} = \mathbf{r}_0 + \mathbf{v}t\)

Worked Example: A toy boat starts at position \(\mathbf{r}_0 = (4\mathbf{i} + 5\mathbf{j})\text{ m}\) and travels with constant velocity \(\mathbf{v} = (3\mathbf{i} - 2\mathbf{j})\text{ m s}^{-1}\). Find its position after \(4\text{ seconds}\).
\(\mathbf{r} = (4\mathbf{i} + 5\mathbf{j}) + (3\mathbf{i} - 2\mathbf{j})(4)\)
\(\mathbf{r} = (4\mathbf{i} + 5\mathbf{j}) + (12\mathbf{i} - 8\mathbf{j})\)
\(\mathbf{r} = (4 + 12)\mathbf{i} + (5 - 8)\mathbf{j} = 16\mathbf{i} - 3\mathbf{j}\text{ m}\).

B. Resultant Force and Equilibrium

When several forces act on a body, the single total force is called the resultant force (\(\mathbf{F}_{\text{net}}\) or \(\mathbf{R}\)), found by adding all individual force vectors together:

\(\mathbf{F}_{\text{net}} = \mathbf{F}_1 + \mathbf{F}_2 + \mathbf{F}_3 + \dots\)

If a body is in equilibrium (meaning it is stationary or moving at a steady constant speed in a straight line), the forces are balanced and the resultant force is zero:
\(\mathbf{F}_{\text{net}} = 0\mathbf{i} + 0\mathbf{j} = \mathbf{0}\)

C. Newton's Second Law in Vector Form

Newton's Second Law (\(F = ma\)) applies directly in vector form:
\(\mathbf{F} = m\mathbf{a}\)
where:
• \(\mathbf{F}\) is the resultant force vector in Newtons (\(\text{N}\))
• \(m\) is the mass of the object in kilograms (\(\text{kg}\)) — note that mass is a scalar!
• \(\mathbf{a}\) is the acceleration vector in \(\text{m s}^{-2}\)

Worked Example: A constant resultant force of \(\mathbf{F} = (14\mathbf{i} - 21\mathbf{j})\text{ N}\) acts on a particle of mass \(3.5\text{ kg}\). Find the acceleration vector of the particle and its magnitude.
Step 1 (Find \(\mathbf{a}\)):
\(\mathbf{a} = \frac{\mathbf{F}}{m} = \frac{14\mathbf{i} - 21\mathbf{j}}{3.5}\)
\(\mathbf{a} = \left(\frac{14}{3.5}\right)\mathbf{i} - \left(\frac{21}{3.5}\right)\mathbf{j} = 4\mathbf{i} - 6\mathbf{j}\text{ m s}^{-2}\)
Step 2 (Find magnitude \(|\mathbf{a}|\)):
\(|\mathbf{a}| = \sqrt{4^2 + (-6)^2} = \sqrt{16 + 36} = \sqrt{52} \approx 7.21\text{ m s}^{-2}\) (to \(3\) sig. figs.).

Key Takeaway: \(\mathbf{F} = m\mathbf{a}\) links forces and acceleration directly: simply divide or multiply the components by the mass \(m\).

7. Summary & Quick Review Checklist

Before sitting your exam, make sure you feel confident with each of these core skills:

• Scalar vs Vector: Scalars have magnitude only; vectors have magnitude and direction.
• Components: Convert freely between \(\begin{pmatrix} x \\ y \end{pmatrix}\) and \(x\mathbf{i} + y\mathbf{j}\).
• Magnitude: Calculate length using \(|\mathbf{r}| = \sqrt{x^2 + y^2}\).
• Direction: Find angles using \(\tan \theta = \left|\frac{y}{x}\right|\) and a quick sketch diagram.
• Displacement: Use \(\vec{AB} = \mathbf{r}_B - \mathbf{r}_A\).
• Motion Formula: Use \(\mathbf{r} = \mathbf{r}_0 + \mathbf{v}t\) for constant velocity.
• Newton's Law: Use \(\mathbf{F} = m\mathbf{a}\) to connect force and acceleration.

Quick check

Can you answer these now?

Open each question to check the key ideas from this chapter.

What is the key difference between a scalar quantity and a vector quantity?

A scalar has magnitude (size) only, whereas a vector has both magnitude and direction.

How are 2D vectors written in column form and in unit vector form?

As a column vector \(\begin{pmatrix} x \\ y \end{pmatrix}\) or in unit vector form as \(x\mathbf{i} + y\mathbf{j}\), where \(\mathbf{i}\) and \(\mathbf{j}\) are perpendicular unit vectors.

How do you add, subtract, and multiply vectors by a scalar?

Add or subtract the corresponding horizontal (\(\mathbf{i}\)) and vertical (\(\mathbf{j}\)) components separately; multiply each component individually by the scalar.

How do you calculate the magnitude of a vector \(\mathbf{r} = x\mathbf{i} + y\mathbf{j}\)?

Use Pythagoras' Theorem: \(|\mathbf{r}| = \sqrt{x^2 + y^2}\).

How do you find the direction angle \(\theta\) of a vector \(x\mathbf{i} + y\mathbf{j}\)?

Use trigonometry: \(\theta = \tan^{-1}\left|\frac{y}{x}\right|\), where \(y\) is the vertical component and \(x\) is the horizontal component.

How do you find the displacement vector \(\vec{AB}\) between two points with position vectors \(\mathbf{r}_A\) and \(\mathbf{r}_B\)?

\(\vec{AB} = \vec{OB} - \vec{OA} = \mathbf{r}_B - \mathbf{r}_A\).

What is the formula for the position vector \(\mathbf{r}\) of an object moving with constant velocity \(\mathbf{v}\)?

\(\mathbf{r} = \mathbf{r}_0 + \mathbf{v}t\), where \(\mathbf{r}_0\) is the initial position and \(t\) is time.

What condition must be satisfied for a set of forces acting on a body to be in equilibrium?

The resultant force must equal zero: \(\mathbf{F}_{\text{net}} = \sum \mathbf{F} = \mathbf{0} = 0\mathbf{i} + 0\mathbf{j}\).

How is Newton's Second Law expressed in vector form?

\(\mathbf{F} = m\mathbf{a}\), where resultant force \(\mathbf{F}\) and acceleration \(\mathbf{a}\) are vectors, and mass \(m\) is a scalar.

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