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Vector addition by graphical methods

Key idea: Add O-Level vectors to scale using head-to-tail and parallelogram methods, then state the resultant magnitude and direction clearly.

  • SEC G3 Physics 2027
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Learning objectives

  • Represent a physical quantity with a numerical magnitude and unit
  • Recall the six prescribed SI base quantities and their units
  • Use the prescribed SI prefixes from nano to tera
  • Compare orders of magnitude from a typical atom to the Earth
  • Select and justify measuring instruments by range and precision
  • Distinguish scalar and vector quantities and give examples
  • Add two vectors graphically to determine a resultant

1. Definitions

Vector

A vector is a quantity with magnitude and direction.

Resultant

The resultant is the single vector that has the same effect as two or more vectors added together.

Prerequisite

Quick refresher: Scalar & Vector Quantities

What you need for this course

You may be asked to add two vectors and determine the resultant by a graphical method (draw to scale, then measure).

2. Key Ideas

  • A vector is drawn as an arrow: length represents magnitude, arrowhead represents direction.
  • For graphical methods, you must choose a scale (e.g. 1 cm : 2 N) and show it on your diagram.
  • The resultant must be stated as magnitude + direction (with units and a direction phrase).
  • The order does not matter: vector A + vector B = vector B + vector A (same resultant).
  • When you quote an angle, state the reference direction (e.g. “37° north of east”).
  • Convert your measured length (cm) back to the real magnitude using the scale.

3. Detailed Explanations

Methods overview

MethodHow you drawResultant arrowCommon mistakes
Head-to-tail (triangle)draw vector B from the head of vector Afrom tail of vector A to head of vector Bdrawing from the wrong point, wrong direction
Parallelogramdraw both vectors from the same tail, complete parallelogramdiagonal from the common tailforgetting construction lines, measuring wrong diagonal

Head-to-tail (triangle) method

Head-to-tail addition of perpendicular vectorsA 4 metre arrow points east. From its head, a 3 metre arrow points north. The resultant points from the original tail to the final head and is labelled 5 metres at 37 degrees north of east.Head-to-tail method4 m east3 m northresultant: 5 m37° north of east37°startfinishscale: 1 cm : 1 m
Scroll diagram horizontally to read all labels.
Using a scale of 1 cm to 1 m, draw 4 cm east and then 3 cm north. The measured resultant should be about 5 cm at 37° north of east.
  1. Draw the first vector to scale.
  2. From the head of the first vector, draw the second vector to scale (same direction as given).
  3. The resultant is drawn from the tail of the first vector to the head of the second vector.
  4. Measure the resultant length (convert using your scale) and measure the direction (with a protractor if needed).

Mini-example (1–2 lines): Use scale 1 cm : 1 m. Draw 4 cm east, then 3 cm north from the head. The resultant is the arrow from the starting tail to the final head.

Parallelogram method

Parallelogram addition of two equal vectorsTwo 5 newton arrows share a tail. One points east and one points 60 degrees north of east. Dashed parallel lines complete a parallelogram, and its diagonal is labelled as the 8.7 newton resultant at 30 degrees north of east.Parallelogram method5 N east5 N at 60°north of eastresultant: 8.7 N30° north of east60°
Scroll diagram horizontally to read all labels.
For two equal 5 N vectors separated by 60°, the parallelogram diagonal gives a resultant of about 8.7 N at 30° north of east.
  1. Draw the two vectors from the same starting point (tail).
  2. Complete the parallelogram.
  3. The diagonal from the common tail gives the resultant.

Mini-example (1–2 lines): Use scale 1 cm : 1 N. Draw 5 cm east and 5 cm at 60° north of east from the same tail. The diagonal gives the resultant.

Measuring magnitude and direction

  1. Measure the resultant length on the diagram using a ruler (in cm).
  2. Convert it back using your scale.
  3. Measure the angle using a protractor and state it clearly (e.g. “20° south of east”).

Mini-example: scale 1 cm : 2 N, measured resultant is 6.2 cm ⇒ magnitude is 12.4 N.

Special case: perpendicular vectors as a check

If the two vectors are at right angles, you can use Pythagoras (and basic trigonometry) to check your measured result:

R = square root of (A² + B²); θ = tan⁻¹ (B/A)

Resultant of two equal vectors depends on angle

A line plot showing how the resultant magnitude changes as the angle between two equal 5 N vectors increases from 0° to 180°.

Scroll across the graph to read all labels.

A line plot showing how the resultant magnitude changes as the angle between two equal 5 N vectors increases from 0° to 180°.A line plot showing how the resultant magnitude changes as the angle between two equal 5 N vectors increases from 0° to 180°.
This visualises what you see in a scale diagram: the resultant gets smaller as the angle opens out (e.g. 0° gives 10 N; 180° gives 0 N).
Open full-size graph
View figure data
Values for Resultant of two equal vectors depends on angle
Angle between vectors (°)A = B = 5 N
010
309.66
608.66
907.07
1205
1502.59
1800

4. Common Mistakes

  • Not drawing vectors to scale (then your measurement is meaningless).
  • Reversing a direction (e.g. drawing east as west).
  • Drawing the second vector from the wrong point in the head-to-tail method.
  • Forgetting to draw arrowheads (then direction is unclear).
  • Not converting your measured cm back to the real magnitude using the scale.
  • Measuring the angle from the wrong reference line (always state “from east”, “from north”, etc.).
  • Reporting only a magnitude without a direction.

5. Exam Tips

  • Always show your scale on the diagram.
  • Label vectors clearly (including directions like east/north).
  • Use a sharp pencil and ruler (messy diagrams lead to wrong measurements).
  • Final answer template: “Resultant = … (unit) at …° … of …”.
  • Quote a sensible number of significant figures based on your measurement (usually 2–3 s.f.).

6. Worked Examples

Modelled example 1

Two perpendicular displacements (head-to-tail)

Core

Problem

A displacement of 4 m east is followed by 3 m north. Find the resultant displacement by a graphical method.
Study the worked solution

  1. Choose and state a scale

    Method

    Use, for example, 1 cm : 1 m.

    Reason

    The drawn arrow lengths must represent the physical magnitudes consistently.

    Working

    4 m → 4 cm and 3 m → 3 cm.

  2. Draw head-to-tail

    Method

    Draw 4 cm east, then 3 cm north from its head.

    Reason

    Successive displacements join head-to-tail in their stated directions.

    Working

    Draw the resultant from the first tail to the final head.

  3. Measure magnitude and direction

    Method

    Measure the resultant arrow and its angle from east.

    Reason

    A vector answer requires both magnitude and referenced direction.

    Working

    Resultant ≈ 5.0 m at 37° north of east.

Guided practice 2

Two equal forces at 60° (parallelogram, non-right-angle)

About 7 min

Problem

Two forces act at a point: 5 N east and 5 N at 60° north of east. Use a parallelogram construction to determine their resultant.

Record measurements from your scale diagram

Unit: N
Unit: degrees

Hints

Hint 1: use a common tail
Draw both 5 N arrows from the same starting point.
Hint 2: choose the correct diagonal
Complete the parallelogram and measure the diagonal from the common tail.
View solution step by step

  1. Construct the parallelogram

    Method

    Use 1 cm : 1 N, draw both 5 cm arrows from one tail, and complete parallel sides.

    Reason

    This construction adds vectors that act from the same point.

    Working

    The resultant is the diagonal starting at the common tail.

  2. Measure and report

    Method

    Convert diagonal length using the scale and state its referenced angle.

    Reason

    Magnitude alone is not a complete force vector.

    Working

    Resultant ≈ 8.7 N at 30° north of east.

Common misconception 3

Opposite directions (1D vectors)

Find and correct the mistake

Learner response

Forces of 10 N east and 6 N west act on an object. A student adds their magnitudes and reports 16 N east. Locate the error.

Carry direction into the sum

Unit: N

View solution step by step

  1. Assign signed components

    Method

    Take east as positive and write the westward force as negative.

    Reason

    Opposite vector directions cannot be added as two positive magnitudes.

    Working

    F_R = 10 + (-6) = 4 N

  2. State the direction

    Method

    Report 4 N east.

    Reason

    The positive resultant follows the chosen east-positive convention.

    Working

    Resultant: 4 N east.

Examiner practice 4

Three vectors (head-to-tail)

4 marks

Examination question

A student walks 6 m east, 4 m north, then 2 m west. Determine the resultant displacement by a graphical method. [4 marks]

Show scale, construction and complete vector result

View solution step by step

  1. Draw all three vectors

    2 marks

    Method

    State 1 cm : 1 m and draw 6 cm east, 4 cm north, then 2 cm west head-to-tail.

    Reason

    The walk is a sequence, so every new vector begins at the previous head.

    Working

    Draw the resultant from the original tail to the final head.

  2. Measure the resultant

    2 marks

    Method

    Measure its length and angle from east.

    Reason

    The final answer must include magnitude, unit and direction.

    Working

    Resultant ≈ 5.7 m at 45° north of east.

Challenge 5

Finding a missing vector (vector subtraction, graphical)

Minimal support

Missing-vector transfer

A resultant displacement vector R is 10 m east and the first displacement vector A is 6 m east. Construct and state the second displacement vector B.

Find the connector between known arrowheads

Hints

Hint 1: use the vector relationship
vector R = vector A + vector B.
Hint 2: construct the missing arrow
Draw vector R and vector A from the same tail; connect the head of vector A to the head of vector R.
View solution step by step

  1. Rearrange the vector relationship

    Method

    Write vector B = vector R- vector A.

    Reason

    The unknown is the vector that completes vector A to reach vector R.

    Working

    B = 10-6 = 4 m

  2. State the constructed direction

    Method

    Draw from A’s head to R’s head and report east.

    Reason

    Both known arrows lie east on the same line and R extends beyond A.

    Working

    vector B = 4 m east

7. Mind Stretchers

Mind stretcher 1: Finding the equilibrant (the “balancing” vector)Extension

Problem 1: Two forces act at a point: 8 N east and 6 N north. Find the single force that will make the resultant force zero.

Show Answer
  • Draw the resultant of the two forces by a graphical method (scale + measurement).
  • The required force (equilibrant) has the same magnitude as the resultant but points in the opposite direction.

Check: resultant is 10 N at 37° north of east, so the equilibrant is 10 N at 37° south of west.

Mind stretcher 2: Converting from your diagram measurementExtension

Problem 2: A student uses a scale of 1 cm : 2 N. The resultant arrow measures 6.2 cm and is directed 20° south of east. State the resultant force (magnitude and direction).

Show Answer
  • Magnitude: 6.2 × 2 = 12.4 N
  • Direction: 20° south of east

Continue with the next resource in this course.

Course and syllabus information
Course
SEC G3 Physics
Edition
SEC G3 Physics 2027