Vectors describe quantities that have both a size AND a direction — like a displacement of 50 m north, or a force of 30 N at 40° above the horizontal. In this chapter you will learn how to break vectors apart into horizontal and vertical components, add them algebraically, and find a single resultant vector that replaces a whole group of forces.
1.1 Vectors and Scalars
Imagine asking two friends for directions. Friend A says 'Walk 3 km.' Friend B says 'Walk 3 km north.' Friend A gave you a SCALAR — just a number with a unit. Friend B gave you a VECTOR — a number with a unit AND a direction. Both answered the same question, but only Friend B gave you enough information to actually get somewhere. This is the core difference between scalars and vectors: vectors carry directional information, scalars do not.
Definition
Vector
A physical quantity that has both magnitude and direction.
Definition
Scalar
A physical quantity that has only magnitude and no direction.
Scalars vs Vectors — Common Examples
| Property | Scalar (magnitude only) | Vector (magnitude + direction) |
|---|---|---|
| Distance / Displacement | Distance: 5 km | Displacement: 5 km north |
| Speed / Velocity | Speed: 60 km·h⁻¹ | Velocity: 60 km·h⁻¹ east |
| Mass / Weight | Mass: 70 kg | Weight: 686 N downward |
| Time / Force | Time: 10 s | Force: 30 N at 40° above horizontal |
| Temperature / Acceleration | Temperature: 25 °C | Acceleration: 9,8 m·s⁻² downward |
In diagrams, a vector is represented by an ARROW. The LENGTH of the arrow is proportional to the magnitude of the vector. The DIRECTION of the arrow (the way the arrowhead points) shows the direction of the vector. A scale is always stated — for example, 1 cm = 10 N.
Exam Tip
EXAM TIP — In any answer about a vector quantity, you MUST state both the magnitude (with units) and the direction. Writing 'F = 50 N' scores no marks for direction. Write 'F = 50 N at 30° above the positive x-axis' (or similar). For scalars, direction is never required.
Watch Out
COMMON MISTAKE — Weight is a VECTOR (it has direction: downward toward the centre of the Earth). Mass is a SCALAR. Never confuse the two. Weight W = mg, and its direction is always downward.
Practice Question
Classify each of the following as a vector or a scalar, and give a reason: (a) a force of 200 N pushing a box to the right, (b) a temperature of 37 °C, (c) a velocity of 15 m·s⁻¹ due south, (d) a mass of 5 kg, (e) an acceleration of 9,8 m·s⁻² downward.
(10 marks)
1.2 Resultant and Equilibrant Vectors
Suppose three tugboats are pulling a ship using ropes in different directions. The ship 'feels' all three forces simultaneously. There is ONE imaginary force that would have exactly the same effect as all three tugboats pulling together. That single force is called the RESULTANT. If you want to keep the ship perfectly still (in equilibrium), you need a fourth force equal in magnitude to the resultant but pointing in the OPPOSITE direction — that is the EQUILIBRANT.
Definition
Resultant vector
The single vector that has the same effect as a combination of vectors acting together.
Definition
Equilibrant
The vector that has the same magnitude as the resultant vector but is in the opposite direction. It is the vector that will balance all other vectors and produce equilibrium.
TAIL-TO-HEAD METHOD (also called the head-to-tail or tip-to-tail method): To add vectors graphically using this method, draw the first vector to scale. Then place the TAIL of the second vector at the HEAD (tip) of the first. Continue placing the tail of each new vector at the head of the previous one. The resultant is drawn from the tail of the first vector to the head of the last vector. Its direction is shown by an arrowhead at the end.
PARALLELOGRAM METHOD: This method is used when two vectors act from the SAME POINT. Draw both vectors to scale from the same starting point (tail-to-tail). Complete the parallelogram by drawing lines parallel to each vector. The DIAGONAL of the parallelogram drawn from the common starting point is the resultant vector.
Note
Both methods give the same resultant — they are just different procedures. The tail-to-head method works for any number of vectors. The parallelogram method is most convenient for exactly two vectors acting at the same point.
Exam Tip
EXAM TIP — In graphical vector addition questions, always: (1) state the scale you are using, (2) use a ruler to draw arrows to the correct length, (3) use a protractor to draw angles accurately, (4) measure the resultant with a ruler and convert back using your scale, (5) state the direction using a bearing or angle from a reference direction.
Worked Example
A hiker walks 40 m east, then 30 m north. Use the tail-to-head method (with a scale of 1 cm = 10 m) to find the resultant displacement. Then verify using Pythagoras.
Given
- Displacement 1: 40 m east
- Displacement 2: 30 m north
- Scale: 1 cm = 10 m
Find
Resultant displacement (magnitude and direction)
Solution
- 1Draw vector 1: 40 m east → 4 cm arrow pointing right.
- 2Place the TAIL of vector 2 at the HEAD of vector 1: draw 30 m north → 3 cm arrow pointing up.
- 3Draw the resultant from the tail of vector 1 to the head of vector 2.
- 4Measure the resultant arrow: it is 5 cm long. Convert: 5 cm × 10 m/cm = 50 m.
- 5Measure the angle with a protractor: θ = 36,9° north of east.
- 6Verify: R = √(40² + 30²) = √(1600 + 900) = √2500 = 50 m ✓
- 7θ = tan⁻¹(30/40) = tan⁻¹(0,75) = 36,9° north of east ✓
Practice Question
A boat travels 60 m due east, then 80 m due south. (a) Draw a tail-to-head diagram using a scale of 1 cm = 20 m. (b) Calculate the magnitude of the resultant displacement using Pythagoras. (c) Calculate the direction of the resultant (angle south of east). (d) State the equilibrant of the resultant displacement.
(8 marks)
1.3 Resolving Vectors into Components
Just as we can ADD vectors to get a resultant, we can do the REVERSE — take one vector and SPLIT it into two perpendicular components (a horizontal component and a vertical component). This is called RESOLVING a vector. Think of a ramp: a ball on a slope experiences gravity (pointing straight down). That one force can be split into a component along the slope (making the ball slide) and a component perpendicular to the slope (pressing the ball into the slope). Each component acts independently.
Definition
Component of a vector
One of two or more vectors in different directions that can be combined to give the original vector. Any vector can be resolved into a horizontal and a vertical component.
Formula
Horizontal component
Fx = horizontal component (N or m), F = magnitude of vector (N or m), θ = angle from the horizontal
SI unit: N (or same unit as F)
Formula
Vertical component
Fy = vertical component (N or m), F = magnitude of vector (N or m), θ = angle from the horizontal
SI unit: N (or same unit as F)
Watch Out
COMMON MISTAKE — The component formulas Fx = F cosθ and Fy = F sinθ are correct ONLY when θ is the angle measured FROM THE HORIZONTAL (x-axis). If θ is given from the vertical (y-axis), the formulas swap: Fx = F sinθ and Fy = F cosθ. Always check which axis θ is measured from before substituting.
Worked Example
A force of 80 N acts at 30° above the horizontal. Calculate (a) the horizontal component and (b) the vertical component of this force.
Given
- F = 80 N
- θ = 30° above the horizontal
Find
(a) Fx (b) Fy
Solution
- 1(a) Fx = F cosθ = 80 × cos30° = 80 × 0,866 = 69,3 N (horizontal, in the direction of motion)
- 2(b) Fy = F sinθ = 80 × sin30° = 80 × 0,5 = 40,0 N (vertical, upward)
Worked Example
A displacement vector has magnitude 100 m at 53° above the horizontal. Resolve this vector into its x- and y-components. Then verify by recalculating the original vector from the components.
Given
- F = 100 m
- θ = 53° above the horizontal
Find
Fx, Fy, and verification
Solution
- 1Fx = F cosθ = 100 × cos53° = 100 × 0,6018 = 60,2 m
- 2Fy = F sinθ = 100 × sin53° = 100 × 0,7986 = 79,9 m
- 3Verification: R = √(Fx² + Fy²) = √(60,2² + 79,9²) = √(3624 + 6384) = √10 008 ≈ 100 m ✓
- 4θ = tan⁻¹(Fy/Fx) = tan⁻¹(79,9/60,2) = tan⁻¹(1,327) ≈ 53° ✓
Exam Tip
EXAM TIP — After resolving, always do a quick check: Fx² + Fy² should equal F². This takes 10 seconds and catches sign errors or angle errors before you lose marks.
Practice Question
A 120 N force acts at 60° above the horizontal. (a) Calculate the horizontal component. (b) Calculate the vertical component. (c) In what direction does each component act? (d) If the object is on a flat surface, which component does work in moving the object horizontally?
(8 marks)
1.4 Adding Vectors Algebraically and Finding the Resultant
The algebraic method is far more precise than graphical methods. The strategy is always the same: (1) Resolve every vector into its x-component and y-component. (2) Add all the x-components together to get a single Σx. (3) Add all the y-components together to get a single Σy. (4) Use Pythagoras to find the magnitude of the resultant: R = √((Σx)² + (Σy)²). (5) Use trigonometry to find the direction: θ = tan⁻¹(Σy / Σx). Treat rightward and upward as positive; leftward and downward as negative.
Formula
Magnitude of resultant
R = magnitude of resultant (N or m), Fx = sum of all x-components, Fy = sum of all y-components
SI unit: N (or same unit as components)
Formula
Direction of resultant
θ = angle of resultant from the horizontal (positive x-axis), Fy = total y-component, Fx = total x-component
SI unit: degrees (°)
Note
SIGN CONVENTION — Always define a positive direction for each axis before starting. The standard convention is: rightward = +x, upward = +y. A vector pointing left has a NEGATIVE x-component; a vector pointing downward has a NEGATIVE y-component. Be consistent throughout the entire calculation.
Worked Example
Three forces act on an object: F₁ = 50 N east, F₂ = 40 N north, F₃ = 30 N west. Find the magnitude and direction of the resultant force.
Given
- F₁ = 50 N east (θ = 0°)
- F₂ = 40 N north (θ = 90°)
- F₃ = 30 N west (θ = 180°)
Find
Magnitude and direction of resultant R
Solution
- 1Resolve each force into x (east–west) and y (north–south) components:
- 2F₁: x = +50 N, y = 0 N
- 3F₂: x = 0 N, y = +40 N
- 4F₃: x = −30 N, y = 0 N
- 5Sum of x-components: Σx = 50 + 0 + (−30) = +20 N
- 6Sum of y-components: Σy = 0 + 40 + 0 = +40 N
- 7Magnitude: R = √(20² + 40²) = √(400 + 1600) = √2000 = 44,7 N
- 8Direction: θ = tan⁻¹(40/20) = tan⁻¹(2,0) = 63,4° north of east
Worked Example
Two forces act on a hook: F₁ = 60 N at 40° above the horizontal (to the right), and F₂ = 80 N at 60° above the horizontal (to the left). Calculate the resultant force.
Given
- F₁ = 60 N at 40° above the positive x-axis
- F₂ = 80 N at 120° from the positive x-axis (60° above the negative x-axis)
Find
Resultant R
Solution
- 1F₁: x = 60 cos40° = 60 × 0,766 = +45,96 N; y = 60 sin40° = 60 × 0,643 = +38,57 N
- 2F₂: x = −80 cos60° = −80 × 0,5 = −40,00 N; y = 80 sin60° = 80 × 0,866 = +69,28 N
- 3Σx = 45,96 + (−40,00) = +5,96 N
- 4Σy = 38,57 + 69,28 = +107,85 N
- 5R = √(5,96² + 107,85²) = √(35,5 + 11631,6) = √11667,1 = 107,97 N ≈ 108 N
- 6θ = tan⁻¹(107,85 / 5,96) = tan⁻¹(18,09) = 86,8° above the horizontal (≈ nearly straight up)
Watch Out
COMMON MISTAKE — When using tan⁻¹ to find the angle, remember that a calculator gives an angle between −90° and +90°. If BOTH Σx and Σy are negative (third quadrant), the calculator answer is wrong — you must add 180°. Always draw a rough sketch of the resultant to confirm your angle is in the correct quadrant.
Practice Question
Two forces act simultaneously on a point: F₁ = 100 N due north and F₂ = 75 N due east. (a) Draw a tail-to-head diagram. (b) Calculate the magnitude of the resultant. (c) Calculate the direction of the resultant (angle east of north). (d) What is the magnitude and direction of the equilibrant?
(8 marks)