4 Lesson 4 of 9 · T.1 Tools of Physics
Scale Diagrams
A scale diagram solves a vector problem by accurate drawing: choose a sensible scale, draw the vectors tip-to-tail with a ruler and protractor, then measure the resultant — especially useful when vectors are not perpendicular, and as an independent check on a calculation.
🎯 Hook – trusting a drawing over a calculator
Before calculators, every surveyor, navigator and engineer who needed a vector sum reached for a ruler and protractor, not an equation. A careful scale drawing of two forces at an awkward angle gives an answer you can see is roughly right — and it still catches mistakes a calculator can't: a sign error, a wrong angle, a resultant pointing the wrong way.
💡 Diagnostic question — why would a scale diagram be more useful than Pythagoras for two forces at 50° to each other? Pythagoras (a²+b²=c²) only works for perpendicular vectors; at any other angle you'd need the cosine rule or components — a scale diagram sidesteps the trigonometry entirely and gives a direct, measurable answer.
📐 Method — the triangle rule, step by step
- Sketch first (freehand) to see roughly how big the diagram will be.
- Choose a scale that fills most of the page and converts easily: 1 cm : 1, 2, 5, 10, 20, 50 … units. Write it on the diagram.
- Draw a reference direction (a north arrow or horizontal line).
- Draw the first vector to scale, with the correct angle, using a ruler and protractor.
- Draw the second vector from the tip of the first, again to scale and angle.
- Draw the resultant from the tail of the first vector to the tip of the last (a different arrowhead style helps).
- Measure the resultant's length (±0.1 cm) and angle (±1°), then convert using the scale.
Parallelogram method: draw both vectors from the same point, complete the parallelogram, and measure the diagonal. Equilibrium (closed polygon): if forces are in equilibrium, drawn tip-to-tail they close; the unknown third force is the closing side, from the tip of the second back to the tail of the first.
Fig 1 — layout for two forces at 60°. The scale is written directly on the diagram.
🧭 Bearings and accuracy
Bearings are measured clockwise from north as three digits: north = 000°, east = 090°, south = 180°, west = 270°. A good scale diagram is accurate to about ±1–2% in magnitude and ±1° in angle. Larger diagrams give smaller percentage uncertainties: ±0.1 cm in a 12 cm line is under 1%, but in a 2 cm line it is 5%.
🔍 Deeper notes — diagram vs calculation, sources of error
Calculation is more precise; a diagram is more visual, catches sign and angle mistakes, and works for any number of vectors without trigonometry. In exams, a scale drawing can confirm a calculation's direction (which quadrant the resultant is in).
Sources of error in drawing: pencil thickness (~0.5 mm), protractor alignment (±0.5–1°), cumulative angle errors when many vectors are chained, and misreading the scale. Chaining n vectors roughly multiplies the drawing uncertainty.
Checking with the cosine rule (extension): for two vectors with angle φ between them (tail to tail), \(R^2=a^2+b^2+2ab\cos\phi\). For vectors drawn tip-to-tail with interior angle α between them, \(R^2=a^2+b^2-2ab\cos\alpha\) (α = 180° − φ). Velocity diagrams follow the same rule: velocity relative to air/water + velocity of air/water = velocity relative to ground.
📊 Graph mastery — precision of a scale diagram
| Graph type | Axes (X, Y) | Shape | What to read |
|---|---|---|---|
| % uncertainty of a measured line vs its length | length · % unc. | 1/x curve, falling steeply then levelling | Longer lines (fixed ±0.1 cm) give much smaller % uncertainty |
| Resultant of 30 N & 40 N vs angle between them | angle · R | Falls from 70 N (0°) to 10 N (180°) | 60.8 N at 60°; 50 N at 90° |
Fig 2 — a fixed ±0.1 cm reading error becomes a smaller percentage as the drawn line gets longer.
✍️ Worked example 1 — two forces at 60°
1
Identify: 30 N (along 0°) and 40 N (at 60°) act at a point — scale 1 cm = 5 N gives arrows of 6.0 cm and 8.0 cm.
2
Measured: the resultant is about 12.2 cm at 35° → ≈ 61 N at 35°.
3
Check by components: \(R_x=30+40\cos60°=50.0\) N; \(R_y=40\sin60°=34.6\) N.
4
Answer: \(R=\sqrt{50.0^2+34.6^2}=\mathbf{60.8\ N}\) at \(\tan^{-1}(34.6/50.0)=\mathbf{34.7^\circ}\) — the diagram and calculation agree within the drawing's precision. The equilibrant (third force for equilibrium) is 61 N at \(34.7°+180°=\mathbf{214.7°}\).
🌍 Real-world: aircraft in wind
An aircraft heads on bearing 045° at 250 km h⁻¹; wind blows south at 60 km h⁻¹. Scale 1 cm = 25 km h⁻¹: measured ground velocity ≈ 8.5 cm on bearing 057° → ≈ 212 km h⁻¹ on 057°. Check by components: east 176.8, north 116.8 → magnitude 211.9 km h⁻¹, bearing 056.5°.
✓ Examiner tip: aim for the longest vector to be at least 8–10 cm, and quote magnitude to 2–3 s.f. and angle to the nearest degree — consistent with the drawing's own precision.
🚀 Going further — digital tools
Geometry software (e.g. GeoGebra) or a spreadsheet of components gives the same answer more precisely than pencil and paper, but the hand-drawn diagram remains the assessed exam skill and the fastest sanity check with nothing but a ruler.
Challenge: a resultant measures 12.2 ± 0.1 cm at scale 1 cm = 5 N. What is the force and its % uncertainty? \(61.0\pm0.5\) N, i.e. about 0.8%.
📋 Quick practice — answers in italics
(a) Choose a suitable scale on A4 paper for vectors of 250 m, 400 m and 120 m. 1 cm = 50 m (giving lines of 5.0, 8.0, 2.4 cm).
(b) By scale drawing, find the resultant of 12 N east and 16 N north. 20 N at 53° north of east (3-4-5 triangle ×4).
(c) State two ways to improve the accuracy of a scale diagram. Use a larger scale (longer lines); use a sharp pencil and a well-aligned protractor.
⚠️ Trap corner – common mistakes
| Misconception | Correct view |
|---|---|
| The scale doesn't need to be written down | No stated scale is a lost mark — always write it on the diagram |
| The second vector starts from the origin | In the triangle method it starts from the tip of the first vector, not the tail |
| Any reference line will do for measuring angles | Measure from the stated reference (horizontal, or north for bearings — clockwise) |
| A small diagram is just as accurate as a large one | Tiny diagrams carry a much larger percentage uncertainty for the same ±0.1 cm reading error |
| Magnitude alone is a complete vector answer | Always state the direction too (angle from a named reference, or bearing) |
⚠️ Most common slip: forgetting to convert the measured length back to the real unit (N, m s⁻¹) using the scale — a beautifully accurate drawing is still wrong if left in centimetres.
state the scale on the diagram
triangle rule: 2nd vector from 1st's tip
bearings: clockwise from north
accuracy ≈ ±1–2% magnitude, ±1° angle
closed polygon ⇒ equilibrium
longer line ⇒ smaller % uncertainty
Practice check — multiple choice
1. The best scale on A4 paper for drawing forces of 45 N and 60 N is:
A 1 cm = 5 N
B 1 cm = 50 N
C 1 cm = 1 N
D 1 cm = 0.5 N
2. With a scale of 1 cm = 20 N, a resultant measures 7.3 cm. The force is:
A 7.3 N
B 27 N
C 146 N
D 2.7 N
3. A line is measured as 7.3 ± 0.1 cm with a scale of 1 cm = 20 N. The uncertainty in the force is:
A ±0.1 N
B ±2 N
C ±20 N
D ±0.2 N
4. In the triangle method, the second vector is drawn:
A from the tail of the first
B parallel to the first
C perpendicular to the first
D from the tip of the first
5. When force vectors drawn tip-to-tail form a closed polygon, it shows that:
A the forces are equal
B the forces are perpendicular
C the resultant force is zero
D the scale is wrong
6. A larger scale diagram is preferred because it:
A reduces the percentage uncertainty in measured lengths
B changes the size of the resultant
C removes the need for a protractor
D makes the vectors perpendicular
7. A scale diagram is most useful, compared with Pythagoras alone, when the vectors are:
A parallel
B not perpendicular
C equal in size
D in opposite directions
8. The direction of a resultant in a scale diagram is found by:
A estimating by eye
B dividing its length by the scale
C using Pythagoras
D measuring the angle with a protractor from a stated reference
9. Two forces of 10 N act at a point with 120° between them. Their resultant is:
A 10 N
B 20 N
C 17 N
D 0 N
10. A scale diagram gives 61 N; the calculation gives 60.8 N. The best conclusion is:
A the diagram is wrong
B the calculation is wrong
C they agree within the precision of the drawing
D the scale is wrong
Practice check — fill in the blanks
Complete the statements:
A scale diagram must always show its written on the page. The triangle rule draws the second vector from the of the first. A closed polygon of force vectors shows the resultant is . Larger diagrams reduce the percentage in the answer. Bearings are measured clockwise from .
Exam‑style questions
Exam question · Construct / Verify
Q1. Forces of 8.0 N and 5.0 N act at a point; the angle between them is 120°. (a) Construct a scale diagram to determine the resultant's magnitude and its angle to the 8.0 N force. State your scale. (b) Verify your answer by calculation using components. [6 marks]
✓ (a) At a stated scale (e.g. 1 cm = 1 N), drawing 8.0 cm at 0° then 5.0 cm at 120° from its tip gives a measured resultant of about 7.0 cm at 38° to the 8.0 N force.
✓ (b) \(F_{2x}=5.0\cos120°=-2.5\), \(F_{2y}=5.0\sin120°=4.33\). \(R_x=8.0-2.5=5.5\), \(R_y=0+4.33=4.33\).
✓ \(R=\sqrt{5.5^2+4.33^2}=\mathbf{7.0\ N}\) at \(\tan^{-1}(4.33/5.5)=\mathbf{38^\circ}\) to the 8.0 N force — matching the diagram.
✓ (b) \(F_{2x}=5.0\cos120°=-2.5\), \(F_{2y}=5.0\sin120°=4.33\). \(R_x=8.0-2.5=5.5\), \(R_y=0+4.33=4.33\).
✓ \(R=\sqrt{5.5^2+4.33^2}=\mathbf{7.0\ N}\) at \(\tan^{-1}(4.33/5.5)=\mathbf{38^\circ}\) to the 8.0 N force — matching the diagram.
Exam question · Draw / Determine / Estimate
Q2. A boat heads due north at 5.0 m s⁻¹ relative to the water; a current flows on bearing 120° at 2.0 m s⁻¹. (a) Draw a velocity vector diagram to scale and determine the boat's velocity relative to the bank (magnitude and bearing). (b) Estimate the uncertainty in your answer from the drawing. [6 marks]
✓ (a) Boat velocity (0, 5.0); current components: east \(=2.0\sin120°=1.73\), north \(=2.0\cos120°=-1.0\).
✓ Resultant: east = 1.73, north = \(5.0-1.0=4.0\); magnitude \(=\sqrt{1.73^2+4.0^2}=\mathbf{4.4\ m\,s^{-1}}\) on bearing \(\tan^{-1}(1.73/4.0)\approx\mathbf{023^\circ}\).
✓ (b) A careful diagram is accurate to about ±1–2% in magnitude (±0.1 m s⁻¹) and ±1° in bearing.
✓ Resultant: east = 1.73, north = \(5.0-1.0=4.0\); magnitude \(=\sqrt{1.73^2+4.0^2}=\mathbf{4.4\ m\,s^{-1}}\) on bearing \(\tan^{-1}(1.73/4.0)\approx\mathbf{023^\circ}\).
✓ (b) A careful diagram is accurate to about ±1–2% in magnitude (±0.1 m s⁻¹) and ±1° in bearing.
Exam question · Compare
Q3. Compare the use of scale diagrams and trigonometric calculation for combining vectors, referring to accuracy, speed and the types of problem each suits. [4 marks]
✓ Accuracy: calculation is exact (limited only by the data); a diagram is limited to about ±1–2% by ruler/protractor precision.
✓ Speed: a quick calculation (especially with components) is usually faster than drawing to scale, especially for many vectors.
✓ Suitability: diagrams are useful when vectors are not perpendicular and trigonometry would be awkward, when a question explicitly asks for a construction, or as a visual check on a calculated direction/quadrant.
✓ Speed: a quick calculation (especially with components) is usually faster than drawing to scale, especially for many vectors.
✓ Suitability: diagrams are useful when vectors are not perpendicular and trigonometry would be awkward, when a question explicitly asks for a construction, or as a visual check on a calculated direction/quadrant.
Exam question · Paper 1B practical skills — how accurate are scale diagrams?
Q4. Five students each construct a scale diagram (1 cm = 5 N) for forces of 30 N at 0° and 40 N at 60°.
(a) Calculate the mean length and angle, and their uncertainties (half the range). (b) Convert the mean length and its uncertainty into newtons. (c) Calculate the resultant by components and compare with the mean diagram value. (d) Calculate the percentage difference between the diagram and calculated values. (e) Suggest two reasons for the spread in results and one improvement. [9 marks]
| Student | Length/cm | Angle/° |
|---|---|---|
| A | 12.1 | 35 |
| B | 12.3 | 34 |
| C | 12.0 | 36 |
| D | 12.4 | 35 |
| E | 12.2 | 34 |
✓ (a) Mean length = 12.2 cm, range 0.4 cm, uncertainty ±0.2 cm. Mean angle = 35°, range 2°, uncertainty ±1°.
✓ (b) \(12.2\times5=\mathbf{61.0\pm1.0\ N}\).
✓ (c) \(R_x=30+40\cos60°=50.0\), \(R_y=40\sin60°=34.64\); \(R=\sqrt{50.0^2+34.64^2}=\mathbf{60.8\ N}\).
✓ (d) \((61.0-60.8)/60.8\times100\approx\mathbf{0.3\%}\).
✓ (e) Parallax error reading the ruler/protractor and pencil-line thickness cause the spread; using a larger-scale diagram (or GeoGebra) would reduce it.
✓ (b) \(12.2\times5=\mathbf{61.0\pm1.0\ N}\).
✓ (c) \(R_x=30+40\cos60°=50.0\), \(R_y=40\sin60°=34.64\); \(R=\sqrt{50.0^2+34.64^2}=\mathbf{60.8\ N}\).
✓ (d) \((61.0-60.8)/60.8\times100\approx\mathbf{0.3\%}\).
✓ (e) Parallax error reading the ruler/protractor and pencil-line thickness cause the spread; using a larger-scale diagram (or GeoGebra) would reduce it.
Exam question · Compare
Q5. Compare the advantages and limitations of scale diagrams and calculation for combining vectors. [3 marks]
✓ Diagrams: visual, catch sign/quadrant errors, handle any number of vectors without trigonometry, but limited to ~1–2% accuracy and are slower for complex problems.
✓ Calculation: exact (within the data's own precision), fast once components are set up, but offers no visual check and a sign or quadrant slip is easy to miss.
✓ Best practice: calculate for the final answer, sketch (or fully construct) a diagram to check the direction and order of magnitude.
✓ Calculation: exact (within the data's own precision), fast once components are set up, but offers no visual check and a sign or quadrant slip is easy to miss.
✓ Best practice: calculate for the final answer, sketch (or fully construct) a diagram to check the direction and order of magnitude.
Exam question · Explain
Q6. Explain how the choice of scale affects the percentage uncertainty of the result, with a numerical example. [2 marks]
✓ A fixed reading uncertainty (e.g. ±0.1 cm) becomes a smaller fraction of a longer line, so choosing a larger scale (longer vectors) reduces the percentage uncertainty.
✓ Example: a 2 cm line with ±0.1 cm carries 5% uncertainty; the same force drawn as a 12 cm line (bigger scale) carries only about 0.8% uncertainty.
✓ Example: a 2 cm line with ±0.1 cm carries 5% uncertainty; the same force drawn as a 12 cm line (bigger scale) carries only about 0.8% uncertainty.
Exam question · Explain
Q7. Explain why errors accumulate when many vectors are drawn tip-to-tail, and how this could be reduced. [2 marks]
✓ Each vector drawn carries its own small length and angle error (pencil thickness, protractor alignment); chaining n vectors tip-to-tail compounds these errors through the whole chain.
✓ This can be reduced by using fewer, longer vectors, by checking the running total periodically with components, or by using geometry software instead of freehand drawing.
✓ This can be reduced by using fewer, longer vectors, by checking the running total periodically with components, or by using geometry software instead of freehand drawing.
Exam question · Explain
Q8. Explain how a closed polygon of force vectors demonstrates equilibrium, and how it lets you find an unknown force. [2 marks]
✓ If the resultant of all the forces is zero, drawing them tip-to-tail returns exactly to the starting point — a closed shape with no gap.
✓ If one force is unknown, draw the known forces tip-to-tail; the segment needed to close the polygon, from the last tip back to the first tail, gives the unknown force's magnitude and direction.
✓ If one force is unknown, draw the known forces tip-to-tail; the segment needed to close the polygon, from the last tip back to the first tail, gives the unknown force's magnitude and direction.
Exam question · Evaluate
Q9. "Scale drawings are obsolete now that we have calculators." Evaluate this claim. [2 marks]
✓ Calculators and computers are faster and more precise for calculation, so a scale diagram is rarely the most efficient way to get a final numerical answer.
✓ However, diagrams remain useful for building geometric intuition, as an independent visual check on a calculated direction, and as an explicitly assessed exam skill ("construct a scale diagram") — so they are not obsolete, but a complement to calculation rather than a replacement.
✓ However, diagrams remain useful for building geometric intuition, as an independent visual check on a calculated direction, and as an explicitly assessed exam skill ("construct a scale diagram") — so they are not obsolete, but a complement to calculation rather than a replacement.
Score: 0 pts