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Choose and state a scale, draw the first force arrow, then draw the second arrow tip-to-tail from its tip. Free GCSE Physics notes and 15 FAQs.
(HT) Vector Diagrams and Resultant Forces is part of Forces and their interactions in GCSE AQA Higher Physics.
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Choose and state a scale, draw the first force arrow, then draw the second arrow tip-to-tail from its tip. Free GCSE Physics notes and 15 FAQs.
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Use this lesson to stay inside Forces and their interactions while connecting the detail back to the wider Forces topic.
Read the core explanations from (HT) Vector Diagrams and Resultant Forces before testing yourself from memory.
Two people push a car. One pushes with 300 N to the right; the other pushes with 300 N to the left. The car does not move. Now imagine both pushing to the right — the car accelerates hard. Same forces, completely different outcome. The difference is direction.
This is why forces are vectors, not just numbers. You cannot add them like ordinary quantities without accounting for the direction each one acts. A force of 300 N east and a force of 300 N west do not give you 600 N — they give you zero.
At Higher Tier, you need to go further than just adding forces along a straight line. When forces act at angles to each other, you need a vector diagram to find the resultant. That is exactly what this lesson covers.
Compare this to a scalar quantity, which has magnitude only. Mass, temperature, and speed are scalars. When you say a car has a mass of 1200 kg, no direction is needed. When you say a force of 500 N acts on it, you must also say which way.
Think of the resultant as the 'net' force — the overall push or pull once everything has been taken into account. If the resultant is large, the object accelerates strongly. If the resultant is zero, the object is in equilibrium.
On a vector diagram, equilibrium looks like a closed shape — the arrows form a loop that returns exactly to the starting point. There is no leftover force in any direction.
Every force can be drawn as an arrow. The length of the arrow represents the magnitude of the force (to a chosen scale), and the direction the arrow points represents the direction the force acts. This is the foundation of every vector diagram you will draw.
Choosing a good scale matters. If you are working with forces of 30 N and 40 N, a scale of 1 cm = 10 N works well — your arrows will be 3 cm and 4 cm long, large enough to measure accurately. A scale that produces arrows smaller than 2 cm will introduce large percentage errors when you measure the resultant.
Always state your scale clearly on the diagram. Examiners award a mark for this, and without it your diagram cannot be interpreted correctly.
When two forces act at an angle to each other, you find the resultant using the tip-to-tail method. Draw the first force arrow. Then, starting from the tip (the arrowhead) of the first arrow, draw the second force arrow. The resultant is the arrow that goes from the tail of the first force to the tip of the second force.
Picture it like two steps on a map. You walk 40 m east, then 30 m north. Your displacement from start to finish is not 70 m — it is 50 m at an angle. Forces work in exactly the same way.
The resultant arrow closes the triangle. Its length (measured with a ruler and converted using your scale) gives the magnitude of the resultant force. The angle it makes with a reference direction (measured with a protractor) gives the direction.
Exam questions give you two forces and the angle between them. Your job is to produce a scale drawing, measure the resultant, and state its magnitude and direction. Here is how to do it reliably every time.
Worked example: Finding the Resultant of Two Forces Using a Scale Drawing
A boat engine exerts a force of 40 N due east. The wind exerts a force of 30 N due north. Use a scale drawing to find the resultant force on the boat, giving its magnitude and direction.
Answer: Resultant force = 50 N at 37° north of east
Notice that the answer has two parts: a magnitude (50 N) and a direction (37° north of east). An answer that gives only the magnitude — '50 N' — is incomplete and will lose the direction mark in an exam.
When three or more forces act on an object that is in equilibrium, something elegant happens on the vector diagram: the arrows form a closed shape. You draw each force tip-to-tail, and the final arrowhead lands exactly back at the starting point. There is no resultant arrow because the resultant is zero.
A classic example is a traffic light hanging from two cables. The weight acts downwards, and the two cable tensions act upwards and outwards at angles. If you draw all three forces tip-to-tail to scale, they form a closed triangle. This tells you the system is in equilibrium — the light hangs still.
You can use this in reverse: if an exam question tells you an object is in equilibrium and gives you two of three forces, you can find the unknown third force by drawing the first two tip-to-tail and then drawing the closing arrow back to the start. The length and direction of that closing arrow give you the missing force.
The tip-to-tail method works in reverse too. Instead of combining two forces into one resultant, you can split a single force into two components — usually a horizontal component and a vertical component. This is called resolving a force.
Imagine a lawnmower being pushed with a force of 50 N at 30° below the horizontal. Only part of that force actually pushes the mower forward along the ground; the rest pushes it downwards into the ground. By resolving the force, you can find exactly how much is doing each job.
On a scale drawing, you represent the 50 N force as a single arrow at 30° to the horizontal. You then draw a horizontal arrow and a vertical arrow that together form the two shorter sides of a right-angled triangle, with the 50 N force as the hypotenuse. Measuring those two sides (and converting with your scale) gives you the two components.
At GCSE, the specification requires scale drawings only — you do not need to use trigonometry to calculate components, though you may find it useful as a check. The marks are awarded for the quality and accuracy of your drawn diagram and your measurements from it.
Vector diagram questions are among the most reliably mark-winning questions on the Higher Tier paper — if you are careful. The marks are awarded for method (drawing the arrows correctly) and accuracy (measuring the resultant correctly). A well-drawn diagram with a small measurement error can still earn most of the marks.
Examiners allow a tolerance of roughly ±2° on angles and ±2 mm on lengths (which translates to a small tolerance on the final force value). This means your diagram does not need to be perfect — but it does need to be drawn with care, using a sharp pencil, a ruler, and a protractor.
The direction of the resultant is the mark that students most often drop. After measuring the length of the resultant arrow and converting it to a force, always measure the angle it makes with a clearly named reference direction. Write something like: '50 N at 37° north of east' or '50 N at 37° above the horizontal'. A bare number like '50 N' will not earn the direction mark.
Published by Revise Right. Use the official specification reference to check the requirements for your course. Content updated: 2026-09-05.
Move to the related lessons or back to the topic page when you need the wider sequence before exam-style practice.
Choose and state a scale, draw the first force arrow, then draw the second arrow tip-to-tail from its tip. Draw the resultant from the tail of the first arrow to the tip of the second, measure its length and angle, and convert using the scale.
The tip-to-tail method finds the resultant of forces acting at an angle: draw the first force arrow, then start the second arrow from the tip of the first. The resultant is the arrow from the tail of the first force to the tip of the second, which closes the triangle.
Measure the resultant arrow with a ruler, convert using the scale, then measure its angle with a protractor. For 40 N east and 30 N north at 1 cm = 10 N, the resultant is 5 cm long, so it is 50 N at 37° north of east.
A closed vector diagram shows that an object is in equilibrium: when all the forces are drawn tip-to-tail, the final arrowhead lands back at the starting point, so the resultant force is zero. A traffic light hanging from two cables gives a closed triangle.
Choose a scale that makes the arrows large enough to measure accurately; for forces of 30 N and 40 N, 1 cm = 10 N gives arrows 3 cm and 4 cm long. Arrows smaller than 2 cm introduce large percentage errors, and the scale must be stated on the diagram.
An unknown third force on an object in equilibrium is found by drawing the two known forces tip-to-tail to scale, then drawing the closing arrow back to the starting point. The length and direction of that closing arrow give the missing force.
A traffic light hanging from two cables has its weight acting downwards and two cable tensions acting upwards and outwards at angles. Drawn tip-to-tail to scale, the three forces form a closed triangle, showing the light is in equilibrium.
A force is resolved with a scale drawing by drawing it as an arrow at the correct angle, then drawing horizontal and vertical arrows that form the two shorter sides of a right-angled triangle, with the force as the hypotenuse. Measuring these sides and converting with the scale gives the components.
Trigonometry is not needed for vector diagrams at GCSE, where scale drawings are used instead, though it can be useful as a check. The marks come from the quality and accuracy of the drawn diagram and measurements.
The direction of a resultant force is stated as an angle measured with a protractor from a clearly named reference direction, such as '50 N at 37° north of east'. A bare '50 N' is incomplete.
Forces of 300 N east and 300 N west give a resultant of zero because force is a vector, with both magnitude and direction. If one person pushes a car with 300 N to the right and another with 300 N to the left, the car does not move; if both push to the right, it accelerates.
Forces of 40 N and 30 N at right angles form the two shorter sides of a 3-4-5 triangle, so the resultant is the hypotenuse, 50 N. With a scale of 1 cm = 10 N, the arrows are 4 cm and 3 cm, and the resultant measures 5 cm.
A force arrow is drawn to scale by making its length represent the magnitude using a chosen scale and pointing it in the direction the force acts. The scale must be stated on the diagram, and it should give arrows longer than 2 cm to keep errors small.
Walking 40 m east then 30 m north gives a displacement of 50 m at an angle, not 70 m. Forces add in exactly the same way: the resultant arrow closes the triangle from the start of the first arrow to the end of the second.
The direction of the resultant is the mark students most often lose in vector diagram questions. After measuring the resultant arrow's length and converting it to a force, its angle from a clearly named reference direction must also be measured and stated.