Pythagoras is brilliant — but only when your two vectors meet at a right angle. The moment they don’t, the neat triangle trick falls apart. So how do you combine a 6 N pull with a 4 N pull at 60° to it? You draw them, carefully, to scale, and simply measure the answer off the page with a ruler and protractor. It feels almost too simple — but for awkward angles it’s exactly what the exam expects.
📘 What you need to know
Use calculation (Pythagoras + trig) when vectors are perpendicular; use a scale drawing when they are not
A scale drawing needs a sharp pencil, a ruler, and a protractor
Choose a sensible scale (e.g. 1 cm = 1 km) so the drawing is large but fits the page
Link vectors head-to-tail, then draw the resultant using the triangle (or parallelogram) method
Measure the resultant’s length with a ruler and its angle (or bearing) with a protractor
Always convert back using the scale — a 5 cm line at 1 cm = 2 km means 10 km
When to draw instead of calculate
There are two routes to a resultant, and the angle between the vectors decides which to take. If they’re at 90°, calculation wins — it’s quick and exact. If they meet at any other angle, a right-angled triangle no longer exists, so you fall back on an accurate scale drawing.
Perpendicular?
→ yes →
Calculate (Pythagoras + trig)
Not perpendicular?
→ so →
Scale drawing (ruler + protractor)
Think of a scale drawing as a physical simulation of the vectors. You’re not calculating the answer — you’re building a little model of the situation and then reading the result off it. That’s why the quality of your drawing is the quality of your answer: a blunt pencil or a wobbly ruler line costs you marks directly.
How to build a scale diagram
The process is the triangle method made physical — precise pencil work replacing the algebra.
🎨 Drawing the diagram in the exam
Choose a sensible scale — big enough to fill the space, small enough to fit; state it (e.g. 1 cm = 1 km)
Draw the first vector to length and direction with a ruler, measuring the angle with a protractor
Link head-to-tail — start the second vector exactly at the head of the first
Draw the resultant from the tail of the first to the head of the last
Measure and convert — ruler for the length (then apply the scale), protractor for the angle or bearing
Draw both forces head-to-tail to scale; the resultant’s length (≈ 8.7 N) and angle are read off with ruler and protractor.
Reading off with the scale
The measurement on your ruler is in centimetres, so the final step is always to translate it back into the real quantity using your chosen scale. Skip this and you’ll hand in a length instead of a force or distance.
Converting back
real value = measured length × scale
Quick recap: perpendicular → calculate; not perpendicular → draw to a stated scale, link head-to-tail, then measure the resultant with a ruler and its direction with a protractor, and convert back.
WE 1
Two ropes pull on a ring. One exerts 6.0 N due east; the other exerts 4.0 N at 60° above the east direction. By making a scale drawing, find the magnitude and direction of the resultant force.
These forces are not perpendicular, so a scale drawing is the right tool. Pick 1 cm = 1 N, draw them head-to-tail, and measure.
Choose scale & draw
1 cm = 1 N → 6.0 cm east, then 4.0 cm at 60°
join head-to-tail, draw resultant from start
Measure & convert
resultant line measures 8.7 cm
R = 8.7 cm × (1 N ÷ cm) = 8.7 N
protractor angle above horizontal ≈ 23°
R = 8.7 N at 23° above eastA component check confirms it: Rx = 6 + 4cos60° = 8.0, Ry = 4sin60° = 3.5, giving 8.7 N at 23°.
WE 2
A yacht sails 6.0 km on a bearing of 020°, then 6.0 km on a bearing of 080°. Use a scale drawing to find the magnitude and bearing of its resultant displacement.
Bearings are measured clockwise from north, so a scale drawing with north pointing up is the clean way to handle them.
With north up, each leg is set with a protractor from north; the resultant is measured back to a bearing.
Draw to scale
1 cm = 1 km, north up
leg 1: 6 cm at 020°, leg 2: 6 cm at 080° head-to-tail
Measure the resultant
length measures 10.4 cm → 10.4 km
bearing from north measures 050°10.4 km on a bearing of 050°The 050° bearing sits exactly halfway between 020° and 080° — a neat check, since the two legs are equal in length.
💡 Top tips
State your scale on the diagram — it earns a mark and stops you forgetting to convert back
Fill the space: a rough rule is that if you could double the scale and still fit, your scale is too small
Measure to 1 mm and 1° — scale-drawing marks come from accuracy, so sharpen that pencil
Right angle? Don’t draw — a perpendicular pair is faster and more precise by calculation
⚠ Common mistakes
Forgetting to convert the measured length back using the scale — reporting centimetres, not the real quantity
Measuring a bearing from the wrong reference — bearings are clockwise from north, not from the horizontal
Drawing too small, so a 1 mm slip becomes a big percentage error
Linking the vectors tail-to-tail for a triangle-method drawing — they must go head-to-tail
Up next: Fundamental & Derived Units in Physics — we leave geometry behind and look at the SI base units every measurement is built from, the prefixes that scale them, and how to derive units like the newton and joule.
Want this to actually click before the exam?
Book a free meeting and let’s work through the tricky bits together.
