Scalars and Vectors
Every physical quantity is one or the other. The distinction takes one sentence to learn and is examined constantly — and it was one of the most-repeated confusions in the whole corpus of lessons behind these notes.
1. The definitions
A scalar has magnitude (size) only. A vector has magnitude AND direction.
Note the spelling: “scalar”, not “scaler”. A recorded error, and worth getting right in a written answer.
| Scalars | Vectors |
|---|---|
| distance | displacement |
| speed | velocity |
| mass | weight |
| time | force |
| energy | acceleration |
| temperature | momentum |
| density, volume, pressure | electric field strength |
The pairs that matter most:
| Scalar | Vector | Difference |
|---|---|---|
| Distance | Displacement | displacement is the straight-line distance in a stated direction |
| Speed | Velocity | velocity has a direction |
| Mass (kg) | Weight (N) | weight is a force, acting downwards |
Force is a VECTOR. Identifying force as a scalar was a specific recorded error — forces have direction, which is exactly why they can cancel out.
Mass is a scalar; weight is a vector. Mass is the amount of matter (kg); weight is the force of gravity on it (N).
A worked distinction. Walk 50 m north, then 50 m south:
- distance travelled = 100 m
- displacement = 0 — you are back where you started
Displacement is zero for any round trip. A recorded error gave 50 m instead of 0; the whole point of displacement is that the direction matters and the two legs cancel.
2. Adding vectors in a straight line
When vectors act along the same line, add them with signs.
Choose a positive direction, then add, treating opposing vectors as negative.
Example: forces of 8 N right and 3 N left.
- Resultant = 8 − 3 = 5 N to the right
Example: 6 N right and 6 N left.
- Resultant = 0 N — the forces are balanced
Always state the DIRECTION of a resultant. “5 N” is an incomplete answer; “5 N to the right” is the full one.
A zero resultant force means constant velocity or rest — not necessarily stationary. Misunderstanding when the resultant force is zero was recorded, and it links directly to Newton’s first law.
3. Vectors at right angles
When two vectors act at 90°, the resultant is the diagonal of the rectangle they form.
By calculation
Magnitude: Pythagoras — R = √(a² + b²) Direction: trigonometry — tan θ = opposite/adjacent
Example: 3 N east and 4 N north.
- R = √(3² + 4²) = √25 = 5 N
- θ = tan⁻¹(4/3) = 53° north of east
Use TANGENT to find the angle from the two perpendicular components — sine and cosine need the hypotenuse. Using sine instead of tangent, and picking the wrong sides for cosine, were both recorded errors.
Identify which angle the question wants, and measure it from the stated reference direction. Getting theta relative to the wrong axis was recorded.
State the direction properly — as an angle from a named direction, or as a three-figure bearing (e.g. 053°) if the question uses bearings.
By scale drawing
- Choose a suitable scale (e.g. 1 cm = 1 N) and write it down
- Draw the first vector to scale, with an arrow
- From its tip, draw the second (tip-to-tail), or complete the parallelogram
- Draw the resultant from the start of the first to the tip of the last
- Measure its length and convert with your scale; measure the angle with a protractor
Choose a scale that fills the space, and state it. A cramped diagram gives an inaccurate answer, and an unstated scale can’t be marked. Misconverting centimetres back into newtons was a recorded error — always convert at the end.
The parallelogram and tip-to-tail methods give the same answer. Use whichever you find clearer, but draw it accurately with a ruler and protractor.
4. Resultant force and motion
The resultant force is the single force that has the same effect as all the forces acting.
- Resultant = 0 → the object stays at rest or continues at constant velocity
- Resultant ≠ 0 → the object accelerates in the direction of the resultant, with F = ma
This is where the scalar/vector distinction earns its keep: forces cancel because they have direction.
5. Mistakes that cost marks
Confusing scalars with vectors.
Calling force a scalar.
Giving a resultant without a direction.
Saying displacement equals distance on a round trip.
Confusing mass with weight.
Adding perpendicular vectors arithmetically — 3 N and 4 N at 90° give 5 N, not 7 N.
Using sine instead of tangent for the angle.
Measuring the angle from the wrong reference.
Not stating the scale on a drawing, or failing to convert back.
Writing a bearing with fewer than three figures.
Frequently asked questions
What is a scalar? A quantity with magnitude only — like speed, mass or energy.
What is a vector? A quantity with magnitude and direction — like velocity, force or displacement.
Is force a scalar or a vector? A vector.
What is the difference between distance and displacement? Distance is how far you travelled; displacement is the straight-line distance in a stated direction.
What is the displacement for a round trip? Zero.
What is the difference between mass and weight? Mass (kg) is a scalar amount of matter; weight (N) is the force of gravity on it — a vector.
How do I add two vectors in a line? Add them with signs, treating one direction as negative.
How do I find a resultant at right angles? Pythagoras for the size, tan⁻¹ for the angle.
What does a zero resultant force mean? The object is at rest or moving at constant velocity.
How do I find a resultant by drawing? Draw both vectors to scale, tip-to-tail or as a parallelogram, then measure the resultant and convert.
Quick revision checklist
- I can define scalar and vector
- I can classify all the common quantities
- I know force, velocity, weight and momentum are vectors
- I know the distance/displacement and speed/velocity distinctions
- I know displacement is zero for a round trip
- I can add vectors along a line using signs
- I always give the direction of a resultant
- I can use Pythagoras for perpendicular vectors
- I can use tan⁻¹ for the angle, from the right reference
- I can find a resultant by scale drawing, stating the scale
- I write bearings in three figures
- I know what a zero resultant force implies
These notes cover scalars and vectors in the Cambridge IGCSE Physics (0625) syllabus and are written for Grade 9–11 / Year 10–11 students. They are based on teaching patterns observed across a large set of one-to-one IGCSE Physics lessons, with particular attention to the errors students make most often and the wording examiners reward. Always check the current syllabus and formula list for your own exam series.
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