A scalar quantity has size only. A vector quantity has size and direction. Telling them apart matters across measurement and quantities and again in motion and forces.
Think of “5 km” versus “5 km north”. The first is a scalar description and the second is a vector description.
How do I classify a quantity?
Ask: does it make sense to say which way it points? Forces push in a direction, so force is a vector. Nobody asks which way a mass of 2 kg points, so mass is a scalar.
| Scalars (size only) | Vectors (size and direction) |
|---|---|
| distance | displacement |
| speed | velocity |
| mass | force (including weight) |
| time | acceleration |
| energy | momentum |
| temperature |
Always check your syllabus list on the Cambridge Physics 0625 page for the quantities required in your exam year.
Worked example
A student walks 3 m east and then 4 m north. Find the distance walked and the displacement from the start. (Invented example data.)
Step 1, distance (scalar): add the lengths of the path: 3 + 4 = 7 m.
Step 2, draw the vectors head to tail. The east leg and the north leg are at right angles, so the start-to-finish line is the hypotenuse of a right-angled triangle.
Step 3, size of displacement: √(3² + 4²) = √(9 + 16) = √25 = 5 m.
Step 4, direction: tan θ = 4/3, so θ = 53° (to the nearest degree) north of east. Checking with the triangle, sin 53° is about 0.80, matching 4/5.
Answer: distance 7 m, displacement 5 m at 53° north of east.
The triangle and bearings reasoning board lets you test this kind of right-angled construction with other values.
The mistake to watch for
A common slip is to add magnitudes of vectors that are not in the same direction.
Mistaken answer: displacement = 3 + 4 = 7 m
The student treated displacement like distance. The straight line from start to finish cannot be as long as the path walked around the corner.
The correction is to ask whether the quantity is a vector and, if so, to draw it. Another useful check is that displacement is never longer than the distance travelled along the path. Here 5 m is shorter than 7 m, as it should be.
Check yourself
1. Classify each as scalar or vector: speed, force, time, velocity, energy.
Show answer
Scalars: speed, time, energy. Vectors: force, velocity.
2. A runner completes one full lap of a 400 m track. State the distance and the displacement.
Show answer
Distance = 400 m. The runner ends at the start, so displacement = 0 m.
3. Two forces act on an object at right angles: 6 N east and 8 N north. Find the size of the resultant force.
Show answer
√(6² + 8²) = √(36 + 64) = √100 = 10 N.
Where this leads next
Once you can separate vectors from scalars, you are ready for stating a realistic uncertainty from supplied instrument data, and later for motion graphs and resultant forces. Revisit choosing a unit appropriate to a quantity if units still slip, and test yourself with the measurement and quantities practice set.
If diagrams and directions are where your marks disappear, our teachers can draw the situations with you in online one-to-one Physics tuition.