Vectors & Scalars

What is a Scalar?

A scalar quantity has magnitude (size) only — no direction is needed to describe it fully. Scalars are added and subtracted using ordinary arithmetic.

Scalar quantity	Symbol	SI unit
Distance	d	metre (m)
Speed	v	m·s⁻¹
Mass	m	kilogram (kg)
Temperature	T	kelvin (K) or °C
Time	t	second (s)
Energy	E	joule (J)

What is a Vector?

A vector quantity has both magnitude AND direction. You must always state both to give a complete description of a vector.

Vector quantity	Symbol	SI unit
Displacement	s or x	metre (m)
Velocity	v	m·s⁻¹
Acceleration	a	m·s⁻²
Force	F	newton (N)
Momentum	p	kg·m·s⁻¹
Weight	W	newton (N)

Notation: Vectors are written in bold (e.g. F) or with an arrow above the symbol (→F). The magnitude of a vector is written as |F| or simply F (not bold).

Vector Diagrams

A vector is represented by an arrow. The length of the arrow represents its magnitude (drawn to scale) and the direction the arrow points represents the vector's direction.

Always choose a scale (e.g. 1 cm = 10 N) and a reference direction (e.g. East = positive) before drawing a vector diagram.

Adding Vectors in One Dimension (1D)

In 1D, choose a positive direction convention (e.g. East = positive, right = positive). Vectors in the positive direction get a + sign; vectors in the negative direction get a − sign. The resultant is their algebraic sum.

Example: A person walks 3 m East then 5 m West.
Let East = positive: +3 m + (−5 m) = −2 m
The resultant displacement is 2 m West.

The tail-to-head method is a graphical technique: draw the second vector starting exactly at the tip (head) of the first. The resultant goes from the tail of the first vector to the head of the last.

If vectors form a closed polygon (the head of the last meets the tail of the first), the resultant = 0. This is called a closed vector diagram.

Resolving Vectors into Components (2D)

Any 2D vector F at angle θ to the horizontal can be split into two perpendicular components:

Fₓ = F cos θ (horizontal component)
Fᵧ = F sin θ (vertical component)

where θ is measured from the positive x-axis (horizontal).

Adding 2D Vectors Using Components

To find the resultant of multiple 2D vectors:

Resolve each vector into its x- and y-components.
Add all x-components: ΣFₓ
Add all y-components: ΣFᵧ
Find the resultant magnitude: R = √(ΣFₓ² + ΣFᵧ²)
Find the angle: θ = arctan(ΣFᵧ / ΣFₓ)

R = √(ΣFₓ² + ΣFᵧ²)
θ = arctan(ΣFᵧ / ΣFₓ)

Worked example: Vector A = 5 N at 0° (East); Vector B = 4 N at 90° (North).
Aₓ = 5 cos 0° = 5 N; Aᵧ = 5 sin 0° = 0 N
Bₓ = 4 cos 90° = 0 N; Bᵧ = 4 sin 90° = 4 N
ΣFₓ = 5 N; ΣFᵧ = 4 N
R = √(5² + 4²) = √41 ≈ 6.40 N
θ = arctan(4/5) ≈ 38.7° North of East

IEB Extension — Unit Vectors & Vector Subtraction

In IEB you will use unit vector notation: î represents a unit vector in the x-direction; ĵ represents a unit vector in the y-direction. Any 2D vector can be written as F = Fₓî + Fᵧĵ.

Vector subtraction is defined as adding the negative: A − B = A + (−B). The negative of a vector has the same magnitude but opposite direction. This is important when finding changes in velocity: Δv = vf − vi.

3-force equilibrium: When three forces act on an object in equilibrium (net force = 0), their vector sum is zero. Drawn tail-to-head, they form a closed triangle. You can solve for unknown forces using trigonometry or the component method.

Complete these questions in your exercise book. Show all working. Include a neat vector diagram where applicable. Use g = 9.8 m·s⁻² unless stated otherwise.

Question 1

Classify each of the following as a scalar or a vector, and give the SI unit: (a) 50 km/h North, (b) 200 J, (c) 9.8 m·s⁻² downward, (d) 37°C, (e) 500 N to the left.

Question 2

A cyclist travels 8 km East, then 3 km West, then 5 km East. Draw a vector diagram to scale (1 cm = 2 km). Find the total distance travelled and the resultant displacement (magnitude and direction).

Question 3

A force of 12 N acts at 30° above the horizontal. Calculate its horizontal and vertical components. Round to two decimal places.

Question 4

Two forces act on an object: F₁ = 6 N East and F₂ = 8 N North. Using the component method, find the magnitude and direction of the resultant force. Draw the tail-to-head vector diagram.

Question 5

Three vectors are added: P = 4 N at 0°, Q = 3 N at 90°, R = 5 N at 180°. Show whether these vectors form a closed vector diagram or not. Calculate the magnitude and direction of the resultant.

Question 6 — Force Table (three non-axis-aligned forces)

Three ropes pull on a ring, with magnitudes and directions (measured anticlockwise from the positive x-axis) recorded below:

Force	Magnitude (N)	Angle from +x-axis
F₁	12	0°
F₂	9	120°
F₃	15	250°

(a) Calculate the x-component and y-component of each force. Round each to two decimal places.
(b) Calculate ΣFₓ and ΣFᵧ (the sum of all x-components and all y-components).
(c) Calculate the magnitude of the resultant force, R = √(ΣFₓ² + ΣFᵧ²).
(d) Calculate the direction of the resultant, θ = arctan(ΣFᵧ / ΣFₓ), and state which quadrant it lies in (use the signs of ΣFₓ and ΣFᵧ to decide).

What is a Scalar?

What is a Vector?

Vector Diagrams

Adding Vectors in One Dimension (1D)

Resolving Vectors into Components (2D)

Adding 2D Vectors Using Components

Vector Addition Sandbox