Scalars and vectors
Physical quantities are either scalars (size only) or vectors (size and direction).
Scalars
Have magnitude only. Examples:
- Mass (kg).
- Distance (m).
- Speed (m/s).
- Energy (J).
- Temperature (°C/K).
- Time (s).
You add scalars by simple arithmetic — 3 kg + 5 kg = 8 kg.
Vectors
Have magnitude AND direction. Examples:
- Displacement (m, with direction).
- Velocity (m/s, with direction).
- Acceleration (m/s², with direction).
- Force (N).
- Momentum (kg m/s).
- Weight (N — direction is downwards toward Earth's centre).
Adding vectors needs care — you can't simply add magnitudes if they're in different directions.
Speed vs velocity, distance vs displacement
- Distance is the total path length covered (scalar).
- Displacement is the straight-line distance from start to finish, with direction (vector).
- Speed is rate of distance (scalar).
- Velocity is rate of displacement (vector).
Example: walk 5 m east then 5 m back. Distance = 10 m, displacement = 0. Speed (avg) = 10/t, velocity (avg) = 0.
Adding vectors graphically
Draw arrows tip-to-tail. The resultant is from the start of the first arrow to the tip of the last.
- Two co-linear vectors: just add or subtract magnitudes.
- Two perpendicular vectors: use Pythagoras and trig.
✦Worked example
A boat heads north at 4 m/s. The river flows east at 3 m/s. Find the boat's actual velocity.
- Magnitude: $\sqrt{4^2 + 3^2} = 5$ m/s.
- Direction: tan⁻¹(3/4) = 37° east of north.
- Result: 5 m/s at 37° E of N.
⚠Common mistakes
- Treating displacement as a scalar.
- Adding velocities of different directions arithmetically without resolution.
- Confusing average speed with average velocity.
- Forgetting that weight (a force) is a vector — points downwards.
AI-generated · claude-opus-4-7 · v3-deep-physics