Pythagoras and trigonometry only work neatly when vectors meet at right angles. But real problems often have vectors at awkward angles — a plane in a slanting wind, two ropes pulling at 50°. When that happens, you fall back on a scale diagram: draw the vectors accurately to a chosen scale, then measure the resultant with a ruler and protractor. It’s low-tech, but it always works.
📚 What you need to know
Use calculation when vectors are perpendicular; use a scale drawing when they’re not
Scale drawings need a sharp pencil, a ruler, and a protractor
Choose a sensible scale (e.g. 1 cm = 1 km) that fills the page
Draw vectors head-to-tail, then draw the resultant from start to finish
Measure the resultant’s length (then convert with the scale) and its angle
Always convert back to the real units using your scale
When to use a scale diagram
There are two ways to combine or resolve vectors, and picking the right one saves time:
Calculation — when the vectors are perpendicular. Pythagoras and trig give an exact answer quickly.
Scale drawing — when the vectors are not at right angles. You draw them accurately and measure.
A quick judgement call before you start: if the vectors are at 90°, don’t waste time drawing — calculate. If they’re at some other angle, reach for the ruler and protractor. Exam scale-diagram questions almost always involve a non-right-angled triangle precisely because that’s when drawing is the sensible method.
The method, step by step
A good scale diagram is really just careful drawing. Follow the same routine every time:
⚛ Drawing a scale diagram
Choose a scale that makes the drawing large (e.g. 1 cm = 1 km). Bigger drawings are more accurate.
Draw each vector to scale with a ruler, at the correct angle with a protractor, linking them head-to-tail.
Draw the resultant from the tail of the first vector to the head of the last.
Measure the resultant’s length with the ruler and convert using your scale.
Measure the angle of the resultant with the protractor.
Draw both legs to scale head-to-tail on the grid, then measure the resultant’s length (× scale) and its angle with a protractor.
Reading the scale correctly
The whole method hinges on the scale. A ruler can only measure a few centimetres, so you shrink real distances down to fit the page — then blow the answer back up at the end.
For example, with a scale of 1 cm = 2 km, a resultant that measures 5 cm on your ruler represents 10 km in the real scenario. Forgetting this final conversion is one of the most common ways to lose marks on an otherwise perfect drawing.
WE 1
A hiker walks 8.0 km due east, then 6.0 km in a direction 45° north of east. By scale diagram (or calculation to check), find the magnitude and direction of their displacement.
Step 1 — choose a scale
Use 1 cm = 1 km, so 8.0 cm then 6.0 cm on paper.
Step 2 — measure the resultant length
The resultant measures about 13.0 cm = 13.0 km.
Step 3 — measure the angle
Protractor gives roughly 19° above the horizontal.
13.0 km at 19° north of eastBecause the two legs aren’t perpendicular, a scale drawing (or component method) is needed — you can’t just use Pythagoras on 8 and 6 here.
Pick a scale that makes the drawing large — small drawings magnify errors.
Use a sharp pencil and measure to about 1 mm.
Always convert the measured length back to real units.
Give both the magnitude and the angle in your answer.
⚠ Common mistakes
Forgetting to convert the ruler measurement back using the scale
Choosing a scale so small the drawing is cramped and inaccurate
Using Pythagoras on non-perpendicular vectors
Measuring the angle from the wrong reference line
Drawing vectors at inaccurate angles (rushing the protractor)
Quick recap: When vectors aren’t perpendicular, use a scale diagram: choose a sensible scale, draw the vectors accurately head-to-tail with a ruler and protractor, draw the resultant from start to finish, then measure its length (convert with the scale) and its angle.
Scale diagrams get you an answer, but every ruler and protractor reading carries its own uncertainty — and those add up. Knowing how to record and combine uncertainties properly is the next essential skill, and it underpins the whole practical side of the course. That’s coming up in Handling Uncertainties.
Scale drawings not adding up?
Book a free meeting and we’ll walk through choosing scales, drawing accurate resultants, and converting back to real units — so those Paper 1B vector questions become easy marks.
