GCSE Physics · Forces and Motion

Adding Vectors in Two Dimensions

In the previous lesson, we added vectors along a single straight line. In this lesson, we are going to look at addition of two-dimensional vectors (2D).

When we are working in two dimensions we can no longer simply add the vectors together — they must be added in a specific way. There are two ways to do this: graphically and by components, in this lesson we are going to look at both. But before we do — let's start by explaining why we can no longer just add the two vectors together.

Why Simple Adding No Longer Works

Suppose a person walks 5 km east and then 10 km north. If we added the values the way we did in the previous lesson, we would get 5 + 10 = 15 km. That is, however, the total distance the person walked — but we are looking for the displacement. We can draw the two distances in a graph - like you can see below, the positive y axis is pointing north and the positive x axis east. The 5.0 km vectors runs from the origin to the right, and the 10 km vector runs from the tip of the first vector straight up. After this walk the person is now 10 km north and 5 km east of the point of origin. The resultant displacement is the vector from the origin to this finishing point, which is not equal to the sum of the individual displacements.

Finding the Resultant Displacement Graphically

The first way to find the resultant displacement is to draw the vectors carefully to scale and then measure it — a ruler for the length and a protractor for the angle. I would strongly encourage you to try this on a piece of paper in front of you. First draw 5 km east, then draw 10 km north, tip-to-tail, taking 1 cm to represent 1 km. Then measure from where you started, the origin, to the endpoint. If you did it correctly, you will find the person is about 11.2 km from the origin. Measuring the angle with a protractor gives about 63° north of east. This is exactly what you need to be able to do for GCSE — you can practise drawing and measuring vector diagrams with the scale-drawing activity below.

But drawing and measuring takes time, and your answer is only as good as your drawing. Where the two vectors are perpendicular (at right angles to each other), there is a more precise mathematical approach: Pythagoras' theorem.

R^{2} = {(5 km)}^{2} + {(10 km)}^{2}

R^{2} = 25 {km}^{2} + 100 {km}^{2} = 125 {km}^{2}

R = \sqrt{125} \approx 11.2 km

Pythagoras gives us the length of the resultant, but not its direction. For the angle we turn to trigonometry. The two vectors and the resultant form a right-angled triangle, so the tangent of the angle θ — measured from east, turning towards north — is the opposite side (the 10 km north) over the adjacent side (the 5 km east):

\tan θ = opposite / adjacent = 10 / 5 = 2

θ = \tan^{- 1} (2) \approx 63.4 °

So the resultant displacement is 11.2 km at about 63.4° north of east — which matches the angle we measured from the scale drawing.

Remember: Pythagoras and this tangent calculation only work when the two vectors are perpendicular — exactly the case here.

So the total displacement is 11.2 km, not 15 km. The 15 km is the distance the person travelled. Remember from the first lesson: distance and displacement are not the same thing — one is a vector, the other is a scalar.

Interactive: Adding Two Vectors

Use the + and − buttons to change the x and y components of each vector. The second vector is joined to the first tip-to-tail, and the resultant is drawn from the origin to the tip of the second vector. Watch the calculation update as you change the vectors.

Vector 1: x y

Vector 2: x y

Try it: can you set the two vectors so the resultant disappears completely (R = 0)? Once you manage it, look closely at Vector 1 and Vector 2 — what has to be true about them?

▼ Show Answer (Click to expand)

Practise: finding the resultant by scale drawing

When two forces act on an object at an angle to each other, together they act like a single force called the resultant. One way to find it is a scale drawing: draw each force as an arrow whose length stands for its size, then join them tip to tail — the tail of the second arrow starts at the tip (arrowhead) of the first. The resultant runs from the start of the first arrow to the tip of the last.

Once the diagram is drawn accurately to scale, you can measure the resultant straight off the page: its length gives the magnitude (using your scale) and a protractor gives its direction. Finding the resultant of two forces this way is exactly the skill that exam questions on resultant forces are testing — and the same method works for any vectors, whether forces, displacements or velocities.

Interactive: Draw and Measure — Tip-to-Tail

Each question gives you two forces on a full x–y grid. Work through the four stages below: place the forces tip to tail, draw the resultant, then measure its length and direction. Drag with a mouse or your finger — everything snaps to the grid unless you untick "Snap to grid". A progress circle turns green as you finish each stage. In this diagram, each grid square represents 1 N. Complete all four exercises to finish.

Scale: each grid square represents 1 N.

You are given two forces:

Force 1 =

Force 2 =

Angles are measured anticlockwise from the right. Find the size and direction of the resultant.

Snap to grid

Now practise on paper

This interactive does some of the work for you — it snaps to the grid and hands you the two forces ready-drawn. In the exam it is different: you are given the two forces and a blank grid, and you have to construct each arrow to scale with a ruler and protractor, join them tip to tail, then measure the resultant. Drawing the vectors yourself is the skill the marks are for, so it is well worth practising by hand.

Download the worksheet below, grab a pencil, ruler and protractor, and draw each pair of forces to scale on the grids provided. Take your time lining up the protractor — neat drawing is what earns the marks. Full worked answers are on the last page, so you can check yourself.

Download the practice worksheet (PDF)

Where this fits in your exam specification

Exam board	Topic
AQA GCSE Physics	4.5.1.4 Resultant forces (Higher Tier)
Edexcel GCSE Physics	Topic 2 — Motion and Forces (CP2)
OCR Gateway GCSE Physics	Topic P2 — Forces

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

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