- Classify common physical quantities as scalars or vectors.
- Represent a vector by a directed arrow.
- Calculate or construct the resultant of two perpendicular vectors.
- Explain why vector direction matters in motion and force problems.
2.1 Scalar and vector quantities
A scalar has magnitude only. Distance, speed, time, mass, energy and temperature can be completely described by a number and unit. A vector has magnitude and direction. Displacement, velocity, acceleration, force, weight, momentum and field strength require a direction as well as a size.
The distinction is not a matter of whether the quantity can be negative. Temperature may be below zero but remains a scalar. Direction must have a physical meaning. “5 m s⁻¹” is speed; “5 m s⁻¹ east” is velocity.
2.2 Distance and displacement
Distance is the total length of the path travelled and is always non-negative. Displacement is the straight-line change in position from start to finish in a stated direction. A runner completing one lap of a 400 m track has travelled 400 m but has zero displacement because the final position is the starting position.
The same distinction carries into speed and velocity. Average speed uses total distance divided by total time. Average velocity uses change in displacement divided by total time. A journey can have a substantial average speed but zero average velocity.
2.3 Drawing vectors
A vector arrow points in the vector direction and its length is drawn to a chosen scale. The scale must be written, for example 1 cm represents 2 N. When adding vectors graphically, place the tail of the second vector at the head of the first, then draw the resultant from the original tail to the final head.
For two perpendicular vectors, the magnitude can be found using Pythagoras. The direction can be found using trigonometry or measured from a scale drawing. The direction must be stated relative to a reference, such as “37° north of east”.
2.4 Components and equilibrium idea
Although full component analysis is not heavily emphasised at this level, it is helpful to understand that a sloping vector can be replaced by perpendicular horizontal and vertical effects. Conversely, perpendicular vectors combine to one resultant.
If the resultant force on an object is zero, the forces are balanced. This does not necessarily mean the object is stationary; it may move at constant velocity. A zero resultant means no acceleration.
Worked examples
A boat is pulled by 8 N east and 6 N north. R = √(8² + 6²) = 10 N. tan θ = 6/8, so θ ≈ 37° north of east.
A student walks 30 m east then 30 m west in 40 s. Distance = 60 m; displacement = 0; average speed = 1.5 m s⁻¹; average velocity = 0.
Practical focus
Use a ruler, protractor and a clear scale to add two force vectors. Repeat the construction using a different order. The final resultant should be the same, illustrating that vector addition is independent of order.
Examination guidance
- Label every vector with both size and direction.
- Do not confuse “resultant” with “larger force”; the resultant is the single vector equivalent of all vectors combined.
- For a right-angled triangle, identify which sides correspond to the actual vector directions before using trigonometry.
Check your understanding
- Classify momentum.
- Two equal perpendicular forces of 5 N act on an object. Find the resultant magnitude.
- Can an object move when the resultant force is zero?
- Vector, because it has the direction of velocity.
- √(5²+5²)=7.1 N.
- Yes. It can move in a straight line at constant speed.
