METRONMETHOD. REASON. LEARN.
Measurement and scientific practice · 12 min

Scalars and vectors

Two journeys can cover the same distance but end in different places. Direction matters when describing changes in position or combining forces.

01Why this matters

Two journeys can cover the same distance but end in different places. Direction matters when describing changes in position or combining forces.

02See it

-4-3-2-10123456786 m east, then 2 m west
The two arrows show a journey from zero to six, then back to four. Total distance is eight metres.
start 0 meast +6 mwest −2 mfinish +4 m
Read left to right: start 0 m → east +6 m → west −2 m → finish +4 m. Each panel connects one part of the example.

03Understand it

A scalar has magnitude only. A vector has magnitude and direction.

Distance, time and mass are scalars. Displacement, velocity and force are vectors. In one dimension, choose a positive direction and represent the opposite direction with a negative sign. This sign is a direction convention, not evidence that the vector has a “negative size”.

04Guided example

Walk 6 m east, then 2 m west. Find distance and displacement.

  1. Distance adds the lengths travelled: 6 + 2 = 8 m.
  2. Choose east as positive. The displacements are +6 m and −2 m.
  3. Add signed displacements: +6 − 2 = +4 m, meaning 4 m east.

05Try it

Walk 5 m east then 3 m west. Total distance?
A little hint

Add both journey lengths.

06Practise it

For the same walk, displacement with east positive?
A little hint

Use +5 and −3.

Forces +12 N and −7 N act along one line. Resultant?
A little hint

Add signed forces.

07Use it in the real world

Two people pull a trolley with 10 N east and 4 N west. The resultant force is 6 N east if these are the only horizontal forces.

Go a little deeper

In two dimensions, perpendicular vector components combine using geometry. A 3 m east and 4 m north displacement has magnitude 5 m, not 7 m.

08Quick check

Walk 4 m east then 4 m west. Displacement?
A little hint

Compare start and finish.

09Summary

A scalar has magnitude only. A vector has magnitude and direction. Add signed displacements: +6 − 2 = +4 m, meaning 4 m east.

Before moving on, explain the worked example in your own words. Use the hint on any question you want to revisit.

10Key idea

Δx = xfinal − xinitial

Δx is signed displacement in m; positions are measured from one origin.

xfinal = xinitial + Δx

Find this in the formula library →