Scalar quantities
Definition
Scalar quantity
A scalar quantity has magnitude only.
- The magnitude of a quantity is its size, or numerical value.
- A scalar has no associated direction, because its value on its own describes it fully.
- Examples of scalars include mass (5 kg5\ \text{kg}5 kg), time (20 s20\ \text{s}20 s), temperature (18 ∘C18\ ^\circ\text{C}18 ∘C), speed (12 m/s12\ \text{m/s}12 m/s), distance and energy.
- Each is complete without a direction: it would be meaningless to describe a mass as “north” or a temperature as “to the left”.
Vector quantities
Definition
Vector quantity
A vector quantity has both a magnitude and an associated direction.
- The direction matters: two vectors with the same magnitude but different directions are different vectors.
- A force of 10 N10\ \text{N}10 N to the right is not the same as a force of 10 N10\ \text{N}10 N to the left.
- Examples of vectors include force, velocity, acceleration, displacement, momentum and weight.
- A complete description gives both magnitude and direction: “a force of 15 N15\ \text{N}15 N” gives only the magnitude, but “a force of 15 N15\ \text{N}15 N upwards” gives both.
Common Mistake
- Do not assume that every quantity with a number and a unit is a scalar.
- Both scalars and vectors have magnitudes and units; only a vector also has a direction.
- Do not confuse speed (a scalar) with velocity (a vector, because it includes direction).
Representing vectors with arrows
- A vector can be represented by an arrow drawn to a chosen scale.
- The length of the arrow represents the magnitude of the vector.
- The direction the arrow points represents the direction of the vector, and the arrowhead shows which way it acts.
- A longer arrow represents a greater magnitude, provided the same scale is used throughout.
- For example, a force to the right is drawn as a horizontal arrow pointing right; a larger force in the same direction is a longer arrow pointing right.
Example
A scale diagram uses the scale 1 cm=5 N1\ \text{cm} = 5\ \text{N}1 cm=5 N. An arrow is 4 cm4\ \text{cm}4 cm long and points to the left. Determine the force it represents.
- Write the scale: 1 cm=5 N1\ \text{cm} = 5\ \text{N}1 cm=5 N.
- Multiply the arrow length by the scale: magnitude=4 cm×5 N/cm\text{magnitude} = 4\ \text{cm} \times 5\ \text{N/cm}magnitude=4 cm×5 N/cm.
- Calculate the magnitude: magnitude=20 N\text{magnitude} = 20\ \text{N}magnitude=20 N.
- Add the direction shown by the arrow: the vector represents a force of 20 N20\ \text{N}20 N to the left.
Describing a vector fully
- To describe how an arrow represents a vector, give both marking points: the length shows the magnitude, and the arrowhead shows the direction.
- To describe a vector itself, always state both its magnitude and its direction; a value and unit alone may not score full marks.
- This is why 10 N10\ \text{N}10 N east and 10 N10\ \text{N}10 N west are different vectors: they have equal magnitude but opposite direction.
Self review
- What does the magnitude of a quantity mean?
- What is the difference between a scalar and a vector quantity?
- What does the length of a vector arrow represent?
- What does the arrowhead on a vector arrow show?
- Why are 10 N10\ \text{N}10 N east and 10 N10\ \text{N}10 N west different vectors?
