Scalars and Vectors

Physics deals with quantities that describe the world around us — speed, force, mass, temperature, velocity. But not all quantities behave in the same way. Some have only a size (magnitude), while others have both a size and a direction. Understanding this distinction is one of the most fundamental skills in mechanics and underpins every topic that follows.

Scalar Quantities

A scalar is a quantity that has magnitude only. It tells you how much of something there is, but nothing about direction.

Quantity	SI Unit	Typical Values
Mass	kg	Electron: 9.11 x 10⁻³¹ kg; Human: 70 kg; Earth: 5.97 x 10²⁴ kg
Speed	m/s	Walking: 1.5 m/s; Sound in air: 343 m/s; Light: 3.0 x 10⁸ m/s
Distance	m	Atom diameter: ~10⁻¹⁰ m; Marathon: 42 195 m
Time	s	Heartbeat: ~1 s; Year: 3.16 x 10⁷ s
Energy	J	Photon of light: ~10⁻¹⁹ J; 1 kWh = 3.6 x 10⁶ J
Temperature	K	Absolute zero: 0 K; Room temp: ~293 K; Sun surface: 5778 K
Power	W	LED bulb: 10 W; Human body: ~100 W; Car engine: ~75 000 W

When you say "the temperature is 293 K" or "the mass is 5.0 kg," direction is irrelevant. A 5.0 kg bag of flour has the same mass regardless of which way it faces.

Scalars are combined using ordinary arithmetic. If you walk 3.0 m and then 4.0 m, the total distance is always 7.0 m, regardless of direction.

Vector Quantities

A vector is a quantity that has both magnitude and direction. You cannot fully describe a vector without stating both.

Quantity	SI Unit	Example
Displacement	m	5.0 m due north
Velocity	m/s	13 m/s at 30° above horizontal
Acceleration	m/s²	9.81 m/s² downward (free fall)
Force	N	50 N at 45° to the surface
Momentum	kg m/s	18 000 kg m/s eastward
Weight	N	687 N downward

Consider the difference between speed and velocity. Speed tells you how fast you are moving (e.g. 5.0 m/s). Velocity tells you how fast and in which direction (e.g. 5.0 m/s due north). Similarly, distance is a scalar (total path length), while displacement is a vector (straight-line distance in a particular direction from start to finish).

Weight vs mass is another important distinction. Mass is a scalar — it is the amount of matter in an object (measured in kg). Weight is a vector — it is the gravitational force acting on the object (measured in N), and it always acts downwards towards the centre of the Earth. Weight = mg, where g = 9.81 m/s².

Reference Table: Gravitational Field Strength

Location	g (m/s²)	Notes
Earth’s surface (standard)	9.81	Used in most exam questions
Equator	9.78	Slightly lower due to rotation
Poles	9.83	Slightly higher, closer to centre
Moon surface	1.62	About 1/6 of Earth
Mars surface	3.72	About 38% of Earth
Jupiter surface	24.8	About 2.5 times Earth

Representing Vectors

Vectors are represented by arrows. The length of the arrow represents the magnitude, and the direction of the arrow represents the direction of the vector. In written work, vectors are often shown in bold (e.g. F) or with an arrow above the symbol.

flowchart LR
    A["Is the quantity fully described by a number and unit alone?"]
    A -->|Yes| B["SCALAR\nExamples: mass, speed,\ndistance, energy, time"]
    A -->|No, direction is also needed| C["VECTOR\nExamples: displacement, velocity,\nacceleration, force, momentum"]

Adding Vectors: The Triangle Rule

Scalars add algebraically, but vectors must be added geometrically because direction matters.

If you walk 3.0 m east and then 4.0 m north, your total distance is 7.0 m (scalar), but your displacement is only 5.0 m at an angle (vector).

The triangle rule for adding two vectors A and B:

Draw vector A.
From the tip of A, draw vector B.
The resultant vector R is the arrow from the tail of A to the tip of B.

Worked Example 1

A boat crosses a river at 4.0 m/s due east while the current flows at 3.0 m/s due south. Find the resultant velocity.

The two velocities are perpendicular, so we use Pythagoras:

R = sqrt(4.0² + 3.0²) = sqrt(16 + 9) = sqrt(25) = 5.0 m/s

The angle below the eastward direction:

theta = arctan(3.0 / 4.0) = arctan(0.75) = 36.9°

The resultant velocity is 5.0 m/s at 36.9° south of east (or equivalently, bearing 127°).

Worked Example 2

Two forces act on a point: 8.0 N at 0° (horizontal right) and 6.0 N at 90° (vertically up). Find the resultant force.

R = sqrt(8.0² + 6.0²) = sqrt(64 + 36) = sqrt(100) = 10.0 N

theta = arctan(6.0 / 8.0) = arctan(0.75) = 36.9° above horizontal

The resultant is 10.0 N at 36.9° above the horizontal.

Adding Vectors: The Parallelogram Rule

When two vectors act from the same point, you can use the parallelogram rule:

Draw both vectors from the same starting point.
Complete the parallelogram by drawing lines parallel to each vector.
The diagonal of the parallelogram from the starting point is the resultant.

This gives the same result as the triangle rule — it is simply a different geometric construction.

Non-Perpendicular Vectors: Using the Cosine Rule

When two vectors are not at right angles, you cannot use simple Pythagoras. Instead, use the cosine rule:

R² = A² + B² - 2AB cos(C)

where C is the angle between the two vectors when drawn tip-to-tail, and R is the resultant.

Worked Example 3

Two forces of 5.0 N and 7.0 N act on an object at 120° to each other. Find the resultant.

When drawn tip-to-tail, the angle inside the triangle is 180° - 120° = 60°.

R² = 5.0² + 7.0² - 2(5.0)(7.0)cos(60°) = 25 + 49 - 70(0.500) = 74 - 35 = 39

R = sqrt(39) = 6.2 N

Resolving Vectors into Components

Any vector can be split into two perpendicular components, usually horizontal and vertical. This is called resolving a vector.

For a vector F at angle theta to the horizontal:

Horizontal component: Fx = F cos(theta)
Vertical component: Fy = F sin(theta)

flowchart TD
    A["Vector F at angle θ to horizontal"] --> B["Horizontal component\nFx = F cos θ"]
    A --> C["Vertical component\nFy = F sin θ"]
    B --> D["Check: F² = Fx² + Fy²"]
    C --> D

Worked Example 4

A force of 50 N acts at 30° above the horizontal. Find the horizontal and vertical components.

Fx = 50 cos(30°) = 50 x 0.866 = 43.3 N (horizontal) Fy = 50 sin(30°) = 50 x 0.500 = 25.0 N (vertical)

Check: sqrt(43.3² + 25.0²) = sqrt(1875 + 625) = sqrt(2500) = 50 N. Correct.

Worked Example 5

A rope pulls a sledge with a force of 120 N at 25° above the horizontal. What is the horizontal pulling force and the vertical lifting force?

Horizontal: 120 cos(25°) = 120 x 0.906 = 108.7 N Vertical: 120 sin(25°) = 120 x 0.423 = 50.7 N

This technique is essential throughout mechanics. Whenever forces act at angles, resolve them into components before applying Newton’s laws.

Equilibrium of Forces

An object is in equilibrium when the resultant force acting on it is zero. This means:

The sum of all horizontal components = 0
The sum of all vertical components = 0

If three forces act on an object in equilibrium, they can be represented as a closed triangle when drawn tip-to-tail. This is because the resultant of any two of the forces is equal and opposite to the third.

Worked Example 6

A 2.0 kg lamp hangs from two cables. Cable A makes an angle of 40° to the horizontal and cable B makes an angle of 60° to the horizontal. Find the tension in each cable.

Weight of lamp: W = mg = 2.0 x 9.81 = 19.6 N (downwards)

Resolving horizontally (taking rightward as positive):

T_A cos(40°) = T_B cos(60°)

Resolving vertically (taking upward as positive):

T_A sin(40°) + T_B sin(60°) = 19.6

From the horizontal equation: T_A = T_B cos(60°) / cos(40°) = T_B x 0.500 / 0.766 = 0.653 T_B

Substituting into the vertical equation:

0.653 T_B x sin(40°) + T_B sin(60°) = 19.6 0.653 T_B x 0.643 + T_B x 0.866 = 19.6 0.420 T_B + 0.866 T_B = 19.6 1.286 T_B = 19.6 T_B = 15.2 N

T_A = 0.653 x 15.2 = 9.9 N

Common Mistakes and Misconceptions

Adding vectors algebraically: Students often add 3.0 m/s north and 4.0 m/s east to get 7.0 m/s. This is the scalar sum (distance-style addition). The vector sum (displacement-style) requires Pythagoras: 5.0 m/s.
Confusing mass and weight: Mass is measured in kg and is a scalar. Weight is measured in N and is a vector. On the Moon your mass is the same but your weight is about 1/6 of that on Earth.
Forgetting direction for vectors: Stating "velocity = 5.0 m/s" is incomplete. Always include the direction.
Resolving errors: Using sine when cosine is needed, or vice versa. Remember: the component adjacent to the angle uses cosine; the component opposite the angle uses sine.
Sign errors in components: When resolving in 2D, forces pointing left or downward must be given a negative value if right/up is chosen as positive.

Exam Tips for Edexcel Physics

When a question asks you to "show that" a resultant has a certain value, you must show full working with the Pythagoras or cosine rule — simply stating the answer earns no marks.
In free-body diagram questions, label every force clearly with its name (not just "F") and indicate its direction with an arrow.
Vector diagrams should be drawn with a ruler and the angles labelled. In "determine" questions, a scale drawing may be acceptable if the question specifies it.
Bearings are measured clockwise from north. Convert to standard angles if needed for trigonometry.
The Edexcel data sheet gives g = 9.81 m/s². Use this value unless told otherwise.

Key Points

Scalars have magnitude only; vectors have magnitude and direction.
Speed is a scalar; velocity is a vector. Distance is a scalar; displacement is a vector.
Vectors are added using the triangle or parallelogram rule, not simple arithmetic.
For non-perpendicular vectors, use the cosine rule.
Any vector can be resolved into perpendicular components using sine and cosine.
An object is in equilibrium when the resultant force is zero in all directions.
Always check your resolved components satisfy F² = Fx² + Fy².