Vectors and scalars

A scalar has only a size (like 20 m/s or 5 kg). A vector has a size and a direction (like 20 m/s north). Forces, velocities and displacements are vectors, so we add them by drawing arrows or by using Pythagoras and trigonometry.

Key facts

  • Scalars: distance, speed, mass, time, energy, temperature, power.
  • Vectors: displacement, velocity, acceleration, force, weight, momentum.
  • Vectors in the same line add or subtract; vectors at right angles combine with Pythagoras.
  • Direction of a resultant: use tan θ = opposite ÷ adjacent (or a bearing from north).
  • Resolving: a vector F at angle θ to the horizontal has components F cos θ (horizontal) and F sin θ (vertical).
  • On a slope at angle θ, weight splits into mg sin θ down the slope and mg cos θ into the slope.

Worked examples

Example 1

A dog runs 40 m east then 15 m west. Find its distance travelled and its displacement.

  1. Then it runs 15 m back the other way.
  2. Distance is a scalar: add up every metre run. 40 + 15 = 55 m
  3. Displacement is a vector: start to finish, in a straight line. 40 − 15 = 25 m east

Answer: distance 55 m; displacement 25 m east

Example 2

A hiker walks 6 km north then 8 km east. Find the size and bearing of her displacement.

  1. The legs are at right angles: the displacement is the hypotenuse. d = √(6² + 8²) = 10 km
  2. The angle from north: tan θ = 8 ÷ 6, θ = 53.1°
  3. A bearing is three figures, clockwise from north. bearing 053°

Answer: 10 km on a bearing of 053°

Example 3

A 60 kg skier stands on a 20° slope. Find the components of her weight along and perpendicular to the slope. (g = 9.8 N/kg)

  1. Her weight: W = 60 × 9.8 = 588 N
  2. Component along the slope (downhill): 588 sin 20° = 201.1 N
  3. Component into the slope. The angle between W and this component is also 20°. 588 cos 20° = 552.5 N

Answer: 201 N down the slope; 553 N into it