Vectors & Scalars — Quick Summary

Mr Toogood's Physics · Mechanics

AQA 3.4.1.1
FR = √(F₁²+F₂²)
Right-angle addition
θ = tan⁻¹(opp/adj)
Resultant angle
H = S cosθ
Horizontal component
V = S sinθ
Vertical component

Scalars vs. vectors

ScalarVector
speed, mass, energyvelocity, force, momentum
distance, time, powerdisplacement, moment, field strength
  • Scalar: magnitude only.
  • Vector: magnitude and direction — shown as +/−, an angle from a reference line, or a compass point.
Distance vs. displacement: distance is the total path length; displacement is the straight line from start to end. A full circular lap has distance = πd but displacement = 0.

Adding vectors in 1D

For vectors acting along the same line, assign consistent + and − directions, then add algebraically. The sign of the resultant tells you its direction — it doesn't matter which direction you call positive, as long as you're consistent throughout.

Vectors at right angles

Adding two vectors at right angles using Pythagoras' theorem

The resultant is the hypotenuse of the right-angled triangle formed by the two vectors.

Drawn nose-to-tail, two perpendicular vectors form a right-angled triangle with the resultant as the hypotenuse:

FR = √(F₁²+F₂²)    θ = tan⁻¹(opposite/adjacent)

Resolving a vector into components

Resolving a single vector into horizontal and vertical components

The reverse process: splitting one vector into two perpendicular components.

Any vector can be split into two components at right angles using trigonometry — essential for forces on inclined planes and projectile motion:

H = S cosθ  (adjacent)    V = S sinθ  (opposite)

Scale diagrams (non-right-angle vectors)

Completed parallelogram of forces construction for a scale diagram

The completed parallelogram — resultant is the diagonal from the origin.

For vectors at any other angle, draw a parallelogram of forces:

  • Draw both vectors to scale from a common origin.
  • Measure the angle between them and complete the parallelogram.
  • Draw the diagonal from the origin — this is the resultant. Measure its length (convert via the scale) and its angle with a protractor.

Getting full marks

You must draw the diagram for full marks in the exam, even if the sine or cosine rule could give the same numerical answer — drawing accurately is a required skill in its own right, not just a means to an answer.

Use a sharp pencil, a ruler and a protractor. Choose as large a scale as reasonably fits the page, to keep measurement errors small.

Exam essentials

Key equations

  • FR = √(F₁²+F₂²)
  • θ = tan⁻¹(opp/adj)
  • H = S cosθ, V = S sinθ

Choosing a method

  • Vectors at 90°: use trigonometry (Pythagoras + tan).
  • Vectors at other angles: scale diagram (or sine/cosine rule, but draw the diagram too if asked).

Common slips

  • Keep the sign convention consistent throughout a 1D addition — don't flip which direction is positive halfway through.
  • Scale diagrams need a sharp pencil, ruler and protractor for marks — a rough sketch isn't enough.
  • Be clear which side (adjacent/opposite) you're resolving relative to before applying sin or cos.