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Learning objectives
Core idea
Key formulae
Worked examples
Explore the graph
Common mistakes
Exam tip
Guided practice
Practice questions
Challenge
Summary
Quick quiz
Same topic, different board
Past paper questions

LearnYMA01WME01 · Mechanics 1Vectors in mechanicsMagnitude, direction and resultant forces

Vectors in mechanics · Subtopic 1 of 2

Magnitude, Direction and Resultant Forces

Writing a vector as $a\mathbf i+b\mathbf j$ splits it into east and north parts. Its magnitude is $\sqrt{a^2+b^2}$ and its direction comes from $\tan\theta=\dfrac ba$ . Add forces part by part to get the resultant, then $\mathbf F=m\mathbf a$ gives the acceleration as a vector.

Edexcel YMA01 WME01≈ 30 minRevision 5 · updated 2026-10-27

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Section 1 of 14

Learning objectives

Find the magnitude of a vector in $\mathbf i,\mathbf j$ form.
Find the angle a vector makes with $\mathbf i$ or $\mathbf j$ .
Add forces to find the resultant.
Use $\mathbf F=m\mathbf a$ and parallel vectors.

Pause and think

Two people pull a box: one with $3$ N east, one with $4$ N north. Does the box feel a $7$ N pull?

Section 2 of 14

Core idea

Start hereTwo tugboats

Two tugs pull a ship, one north-east, one north-west. Which way does the ship go, and how hard is it pulled? Write each force as east and north components, add, and the answer drops out.

The big idea

Forces $a\mathbf i+b\mathbf j$ add component by component. $|\mathbf F|=\sqrt{a^2+b^2}$ ; angle with $\mathbf i$ : $\tan^{-1}\frac ba$ . $\mathbf F=m\mathbf a$ in vector form.

1. Adding force vectors

Adding (3i + j) N and (i + 3j) N head to tail

\mathbf R=4\mathbf i+4\mathbf j

: magnitude

\sqrt{32}=5.66

N, direction

45^\circ

above

\mathbf i

\mathbf R=\sum\mathbf F_k,\qquad|\mathbf R|=\sqrt{R_x^2+R_y^2},\qquad\mathbf R=m\mathbf a

$\mathbf i,\mathbf j$: unit vectors east and north.
$\mathbf R$: resultant.

Worked example $(3\mathbf i+\mathbf j)+(\mathbf i+3\mathbf j)$ N on 2 kg

Resultant, magnitude, direction, acceleration, balancing force.

Goal: Five things.
Strategy: Add components; divide by mass.

$\mathbf R=4\mathbf i+4\mathbf j$
Add.
$|\mathbf R|=\sqrt{32}=5.66\text{ N},\ \tan^{-1}1=45^\circ$
Magnitude, angle with $\mathbf i$ .
$\mathbf a=\tfrac12\mathbf R=2\mathbf i+2\mathbf j$
$\mathbf a=\mathbf R/m$ .
$-4\mathbf i-4\mathbf j$
Equilibrant: equal and opposite.

Answer

5.66

N at

45^\circ

;

\mathbf a=2\mathbf i+2\mathbf j

m s

^{-2}

; balancing

-4\mathbf i-4\mathbf j

✓ Check

|\mathbf a|=2.83=5.66/2

✓.

Worked exampleUnknown components

$(p\mathbf i+2\mathbf j)+(3\mathbf i+q\mathbf j)$ is parallel to $\mathbf i+\mathbf j$ with magnitude $10\sqrt2$ . Find $p$ , $q$ .

Goal: $p$ , $q$ .
Strategy: Parallel to $\mathbf i+\mathbf j$ : equal components.

$(p+3)\mathbf i+(q+2)\mathbf j=k(\mathbf i+\mathbf j)$
Parallel.
$\sqrt{2k^2}=10\sqrt2\Rightarrow k=10$
Magnitude ( $k>0$ for the same direction).
$p=7,\ q=8$
Each component 10.

Answer

p=7

q=8

✓ Check

10\mathbf i+10\mathbf j

has magnitude

10\sqrt2

✓.

Common misconceptionAngle in the wrong quadrant

-2\mathbf i+2\mathbf j

: calculator gives

-45^\circ

; sketch shows

135^\circ

Pause and think

Angle of $(-2\mathbf i+2\mathbf j)$ with $\mathbf i$ ?

Summary

Add components.
$|\mathbf F|=\sqrt{a^2+b^2}$ ; sketch for direction.
$\mathbf F=m\mathbf a$ .

Resultant: Sum of forces.
Equilibrant: Force that balances the resultant.

HistoryThe parallelogram of forces. The idea that two forces combine like the diagonal of a parallelogram appears in the work of Simon Stevin around 1586 and was stated clearly by Newton in the Principia (1687). The

\mathbf i

\mathbf j

notation came much later, from Josiah Willard Gibbs in the 1880s.

Fun factTug of war physics. In a tug of war, a team pulling slightly off the line of the rope wastes part of its effort: only the component along the rope counts. Good teams pull in a straight line.

Section 3 of 14

Key formulae

Magnitude

|a\mathbf i+b\mathbf j|=\sqrt{a^2+b^2}

Direction

\tan\theta=\frac{b}{a}\ \text{(angle with }\mathbf i)

Newton's second law

\mathbf F=m\mathbf a

TrickSketch before tan. Always draw a quick arrow for the vector before using

\tan^{-1}

. Your calculator gives an angle between

-90^\circ

and

90^\circ

; the sketch tells you which quadrant you're really in and which line the angle is measured from.

Section 4 of 14

Worked examples

Example 1 · Resultant0 / 2 steps

Forces

(2\mathbf i+3\mathbf j)

N and

(4\mathbf i-7\mathbf j)

N act on a particle. Find the magnitude of the resultant.

Example 2 · Acceleration0 / 2 steps

Forces

(4\mathbf i-2\mathbf j)

N and

(2\mathbf i+10\mathbf j)

N act on a

2

kg particle. Find the magnitude of its acceleration.

Section 5 of 14

Explore the graph

Adding two forces

Loading the interactive graph…

Try this

The first force is $p\mathbf i+q\mathbf j$ ; the second, $\mathbf i+3\mathbf j$ , is drawn from its tip. The long line is the resultant.
The value is the magnitude of the resultant. Check it with Pythagoras.
Find $p$ and $q$ so the resultant is zero. What does that mean physically?

Forces add tip to tail: add the $\mathbf i$ parts and the $\mathbf j$ parts.

In the real worldSailing into the wind. A sailboat can make progress almost into the wind because the force on the sail and the keel's sideways resistance add up to a resultant pointing forwards. Sailors use vector addition without writing a single

\mathbf i

Section 6 of 14

Common mistakes

Incorrect

|5\mathbf i-12\mathbf j|=5-12

Correct

\sqrt{25+144}=13

Magnitude uses Pythagoras.

IncorrectAngle with the wrong axis

CorrectCheck the sketch

\tan^{-1}\tfrac43

is the angle with

\mathbf i

;

\tan^{-1}\tfrac34

is with

\mathbf j

IncorrectMagnitudes added first

CorrectAdd the vectors, then find the magnitude

|\mathbf a|+|\mathbf b|\ne|\mathbf a+\mathbf b|

IncorrectForgetting to divide by

m

Correct

\mathbf a=\mathbf R/m

\mathbf R

is a force, not an acceleration.

$(3\mathbf i+4\mathbf j)$ N

Magnitude $5$ N, at $53.1^\circ$ to $\mathbf i$ . The 3-4-5 triangle never misses.

Section 7 of 14

Exam tip

Exam tipGive direction as a clear angle. State the angle and what it is measured from, e.g. "

53.1^\circ

above

\mathbf i

". A bare

53.1^\circ

may lose the mark if the reference line isn't clear.

Section 8 of 14

Guided practice

SA-VM-G011 markL1 FoundationCalculatorMagnitude & direction

Find the magnitude of

5\mathbf i-12\mathbf j

Salam Academy Original Practice · Proposed mark scheme

Section 9 of 14

Practice questions

SA-VM1-0011 markL1 FoundationNon-calculatorMagnitude & direction

Find the magnitude of

(-8\mathbf i+6\mathbf j)

Salam Academy Original Practice · Proposed mark scheme

SA-VM1-0022 marksL2 StandardCalculatorMagnitude & direction

Find the angle between the vector

(2\mathbf i+5\mathbf j)

and

\mathbf i

, in degrees to 1 decimal place.

Salam Academy Original Practice · Proposed mark scheme

SA-VM-0012 marksL1 FoundationCalculatorMagnitude & direction

A particle moves with velocity

(3\mathbf i+4\mathbf j)\ \text{m s}^{-1}

. Find the angle between its direction of motion and

\mathbf i

, in degrees to 1 decimal place.

Salam Academy Original Practice · Proposed mark scheme

SA-VM1-0032 marksL1 FoundationNon-calculatorVector forces

Forces

(3\mathbf i+\mathbf j)

(-\mathbf i+4\mathbf j)

N and

(2\mathbf i-2\mathbf j)

N act on a particle. The resultant is

(p\mathbf i+q\mathbf j)

N. Find

p

and

q

Salam Academy Original Practice · Proposed mark scheme

SA-VM-0023 marksL2 StandardCalculatorVector forces

Forces

(2\mathbf i+3\mathbf j)

N and

(4\mathbf i-7\mathbf j)

N act on a particle. Find the magnitude of the resultant, to 3 significant figures.

Salam Academy Original Practice · Proposed mark scheme

SA-VM1-0042 marksL2 StandardNon-calculatorVector forces

A particle of mass

4

kg has acceleration

(2\mathbf i-\mathbf j)

m s⁻². Find the resultant force

(p\mathbf i+q\mathbf j)

N acting on it.

Salam Academy Original Practice · Proposed mark scheme

SA-VM-0053 marksL2 StandardCalculatorVector forces

Forces

(4\mathbf i-2\mathbf j)

N and

(2\mathbf i+10\mathbf j)

N act on a particle of mass

2

kg. Find the magnitude of its acceleration.

Salam Academy Original Practice · Proposed mark scheme

SA-VM-0072 marksL2 StandardCalculatorVector forces

The force

(p\mathbf i+3\mathbf j)

N is parallel to the vector

2\mathbf i+\mathbf j

. Find

p

Salam Academy Original Practice · Proposed mark scheme

SA-VM1-0053 marksL2 StandardNon-calculatorVector forces

Forces

(2\mathbf i+a\mathbf j)

N and

(b\mathbf i-3\mathbf j)

N are in equilibrium with a third force

(5\mathbf i+\mathbf j)

N. Find

a

and

b

Salam Academy Original Practice · Proposed mark scheme

More questions on every skill in this subtopic are in the question bank under Practise.

Section 10 of 14

Challenge

SA-VM-0095 marksL2 StandardCalculatorVector forces

Two forces

(3\mathbf i+5\mathbf j)

N and

(5\mathbf i-11\mathbf j)

N act on a particle of mass

2

kg. Find (i) the magnitude of the acceleration and (ii) the angle the acceleration makes with

\mathbf i

, in degrees to 1 decimal place.

Salam Academy Original Practice · Proposed mark scheme

Section 11 of 14

Summary

$|a\mathbf i+b\mathbf j|=\sqrt{a^2+b^2}$ .
Direction: $\tan\theta=\tfrac ba$ from a sketch.
Resultant: add components.
$\mathbf a=\mathbf F/m$ ; parallel means a multiple.

Section 12 of 14

Quick quiz

Q1 · SA-VM1-0061 markL1 FoundationCalculatorVector forces

Which vector is parallel to

(2\mathbf i-3\mathbf j)

Salam Academy Original Practice · Proposed mark scheme

Section 13 of 14

Same topic, different board

Specification: IAL Mathematics, unit WME01 (Mechanics 1)
Where: M1 · 2 Vectors in mechanics
Expected: Find the magnitude and direction of vectors in i, j form; add forces to find a resultant; use F = ma with vectors; parallel vectors.
Calculator: Calculators are permitted, but exact answers (surds, fractions, π) are expected where the question asks.
Typical style: "Find the angle between the resultant force and the vector i" and "find the magnitude of the acceleration".
Open in: YMA01 · WME01

Section 14 of 14

Past paper questions

3 questions (19 marks) from real exams test this subtopic. Try each one under timed conditions, then mark it with the official mark scheme.

WME01/01 · June 2025Question 47 marks
Two tensions on a barge: an angle, then acceleration against resistance.
Question paper, page 12 Mark scheme
WME01/01 · January 2025Question 18 marks
Acceleration from forces, then a constant that fixes the direction of the resultant.
Question paper, page 2 Mark scheme
WME01/01 · June 2024Question 24 marks
Add force vectors given by magnitude and bearing.
Question paper, page 4 Mark scheme

All YMA01 past paper questions by topic

Salam Academy original content · Content status: reviewed and published. Syllabus mapping: Pearson Edexcel YMA01, 2018 specification (official specification).