GCSE & A-Level Physics · Forces and Motion

Adding Vectors in One Dimension

In the previous lesson, we learned the difference between scalars and vectors: scalars have magnitude only, while vectors have both magnitude and direction.

In this lesson, we will add vectors in one dimension (1D) only — along a single straight line — to find the resultant vector. In the next lesson, we will add vectors in two dimensions (2D).

Before we start let's have a look at the definition of what a resultant vector is.

Resultant vector: A single vector that can replace two or more vectors and have the same effect as all of them acting together. We find it by adding the vectors, so the resultant is also called the vector sum.; What does "the same effect" mean? Imagine a person walks 3 km east and then 2 km east. One single walk of 5 km east would bring them to exactly the same finishing point — so that single walk is the resultant of the two. Replacing several vectors with their resultant changes nothing about the outcome; it just describes it with one vector instead of several.

Interactive: Adding Vectors on a Number Line

The arrow shows a person walking along a straight road. Walking east is positive; walking west is negative. Use the + and − buttons to change the size of the first vector. Then click "Add a second vector" and change its size too. The resultant vector — the single vector with the same overall effect — is drawn underneath the number line.

First vector:

▼ Show Explanation (Click to expand)

Reflection

Using the interactive above, set the first vector to point east (positive) and the second to point west (negative). The person has now walked two stretches of road, yet the resultant vector is shorter than the two walks combined. What is the difference between the total distance the person walked and their displacement? Type your answer and press Enter or click the button below.

Reflection exercises help you pause, think, and connect ideas to real-world examples. Write down what you understand and where you're unsure, then refine your thoughts by checking your response against the feedback.

Let's have a look at some examples, for which we will use distance and displacement.

Distance: How far an object moves in total, along its path. Distance does not involve direction — it is a scalar quantity.
Displacement: The distance from the start point to the finish point measured in a straight line, together with the direction of that line. Displacement is a vector quantity.

Example 1: Walking in the Same Direction

A hiker walks 8 km east, stops for the night, and the next day walks another 6 km east. What is their resultant displacement?

Both vectors point east, so we add them: (+8) + (+6) = +14. The resultant displacement is 14 km east.

Example 2: Walking in Opposite Directions

This time, the hiker walks 8 km east, then turns around and walks 6 km west. West is the negative direction, so the second vector is −6 km.

(+8) + (−6) = +2. The resultant displacement is just 2 km east — even though the hiker walked a total distance of 14 km.

In Example 2, the hiker's distance is 8 + 6 = 14 km, but their displacement is only 2 km east. When we add vectors, we are finding the displacement, not the distance. This distinction matters in physics: it is the reason a runner who completes a full lap of a 400 m track has travelled a distance of 400 m but has a displacement of zero.

Exercise: Adding Vectors in 1D

Click "Add a vector" to place a vector on the number line, then use the + and − buttons to change its size and direction: east is positive, west is negative. Press and hold a button to change the value quickly. You can place up to three vectors. Tick "Show resultant vector" to check the single vector they add up to. Use ↻ to clear the line and start again.

Show resultant vector

Answer the question below, then press Enter or click Check. You can drag the vectors onto the number line above to see the answer visually, but you don't have to — it is there to help. The four circles show your progress: a circle turns green when you answer that type of question correctly. Click "New question" at any time to try different numbers.

Adding vectors along a single line is the foundation for everything that follows. In the next lesson, we will add vectors in two dimensions, where direction can no longer be captured by a simple + or − sign.

Where this fits in your exam specification

Exam board	Topic
AQA GCSE Physics	4.5 Forces
Edexcel GCSE Physics	Topic 2 — Motion and Forces
OCR Gateway GCSE Physics	Topic P2 — Forces
AQA A-Level Physics	3.4 Mechanics and Materials
Edexcel A-Level Physics	Topic 2 — Mechanics
OCR A A-Level Physics	Module 3 — Forces and Motion

Studying a different board or combined science? The content of this lesson is the same — only the topic name differs.

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