Where can I learn Scalars and vectors for Cambridge 9702?

This free lesson on MarkScheme covers syllabus point 1.4 (Scalars and vectors) for Cambridge A-Level Physics. It includes visual revision, key points, and links to real past-paper questions with mark-scheme marking at markscheme.app/courses/9702/1-4-scalars-and-vectors.

Is Scalars and vectors (1.4) on 9702 free to study?

Yes. The Physics 9702 course on MarkScheme is 100% free — including premium visual lessons, exam tips, and past-paper marking. No paywall for Scalars and vectors.

How do I revise Scalars and vectors for 9702 exams?

Start with the visual steps on this page, note the key exam points, then attempt a 9702 past-paper question on Scalars and vectors using MarkScheme's /mark tool. Mark strictly against the official scheme and re-read any marks you lost.

MarkScheme 9702

Opening…

1.4Physics25 min readPREMIUM

Scalars and vectors

Welcome to the world of scalars and vectors! In physics, understanding these two fundamental categories of physical quantities is crucial. Some quantities, like temperature, are simple, needing just a size. Others, like the velocity of a rocket, require both size and a specific direction. Let's dive into how we define, represent, and combine them. This pilot aligns with Senpai Corner AS notes (2025–2027) for topic 1.4.

BY THE END, YOU CAN…

exam checklist — tick these off

1Distinguish between scalar and vector quantities, providing relevant examples.
2Add and resolve vectors using both graphical (scale diagrams) and analytical (component) methods.
3Understand the effect of multiplying a vector by a positive or negative scalar.

STUDY MODE

READING

TYPEFACE

Literata — a reading serif made for e-books; set like a textbook, sits well beside maths.

SIZE

AIRY SPACING

Wider letter and line spacing. The one change with solid evidence for readers who lose their place — worth a try whoever you are.

Pick whatever reads fastest for you — it differs from person to person, by up to a third.

EXPLAIN SIMPLER

The same notes in plain English — no jargon. Key points and exam tips stay.

DIAGRAM NOTES

A small sketch beside each block of notes.

In simple terms

A friendly intro before the formal notes — no formulas yet.

Scalars and vectors

Cambridge 9702 Paper 2 — Scalars and vectors (1.4). Senpai Corner diagram-backed pilot with premium structure and live visuals.

1
1.4 Scalars and vectors.
2
A scalar is a quantity which only has a magnitude.
3
E.g. speed, mass, time and distance.
4
A vector is a quantity with both a magnitude and direction.

What this topic covers

The official Cambridge syllabus points this lesson works through.

1.4.1
Understand the difference between scalar and vector quantities and give examples of scalar and vector quantities included in the syllabus
1.4.2
Add and subtract coplanar vectors
1.4.3
Represent a vector as two perpendicular components

Explore the concept

Use the live diagram, PhET or GeoGebra sim, and synced steps — play it, drag controls, or tap a step.

Figure PhET simulation1 / 4

Loading simulation…

Fetching from PhET…

Vector Addition

Resultant = tip-to-tail sum; components add independently.

If the embed does not load, use Open on PhET.

Simulation by PhET Interactive Simulations, University of Colorado Boulder · Licensed to MarkScheme (public licence CC BY-NC 4.0 since 2026-03-30)

Step-synced diagram — highlights what to look for in the simulation above.

Vectors have magnitude and direction

Vectors have magnitude and direction — show both on diagrams.

26 more simulations for this topic — run them in the Simulations section below

Simulations

Every simulation here runs the real model — try the steps on a card, then check what you see against the notes.

26 simulations · 5 to start with

Start herein this order — each one shows a different piece of the topic

oPhysicsStart here · 19702 1.4 · IB A.1 · IB A.2
Vector Addition
Drag the tips of two vectors; see their resultant by triangle or parallelogram method
Why this one: Triangle and parallelogram methods give the same resultant; watch it shrink as the vectors turn antiparallel.
Try this
1. Drag the two vector tips so they are perpendicular and read the resultant.
2. Switch between the triangle and parallelogram methods for the same pair.
3. Drag one tip until the two vectors point the same way.
Look for Both methods give the same resultant; it is longest when the vectors are parallel and shortest when antiparallel.
Open on oPhysics
Simulation by Tom Walsh, oPhysics.com — made with GeoGebra · Licensed to MarkScheme (site: free for non-profit educational use; applets made with GeoGebra)
The Physics ClassroomStart here · 29702 1.4 · 9702 2.1 · IB A.1
Vector Walk
Three short animated motions; work out the distance travelled and the overall displacement for each
Why this one: Add every step for distance, then draw start-to-finish for displacement: scalar versus vector in one walk.
Try this
1. Add up every step of the first motion to get the distance travelled.
2. Mark the start and finish and find the overall displacement.
3. Repeat for the other two motions and compare the two values each time.
Look for Distance counts every step of the path; displacement is only the straight-line change from start to finish, so it is never larger than distance.
Open on The Physics Classroom
Physics Interactives by The Physics Classroom · Licensed to MarkScheme (site terms otherwise permit linking only)
oPhysicsStart here · 39702 1.4 · IB A.1 · IB A.2
Vector Components
Enter a vector magnitude and direction; see its x and y components graphically and numerically
Why this one: Enter magnitude and angle, read |v| cos θ and |v| sin θ; past 90° the x-component goes negative.
Try this
1. Enter magnitude 10 at 30° and read both components.
2. Keep the magnitude and change the direction to 60°, then 90°.
3. Enter a direction between 90° and 180° and note the sign of the x component.
Look for The components are |v| cos θ and |v| sin θ, so they swap values between 30° and 60° and the x component turns negative past 90°.
Open on oPhysics
Simulation by Tom Walsh, oPhysics.com — made with GeoGebra · Licensed to MarkScheme (site: free for non-profit educational use; applets made with GeoGebra)
The Physics ClassroomStart here · 49702 1.4 · IB A.1
Name that Vector
Twelve challenges: add three of the 25 vectors shown on the grid by components and name the resultant
Why this one: Add x-components and y-components separately and name the resultant: the analytical method drilled.
Try this
1. Read the x- and y-components of each of the three vectors from the grid.
2. Add the x-components and the y-components separately.
3. Name the resultant from the two sums and check it.
Look for The components of the resultant are the sums of the x-components and of the y-components.
Open on The Physics Classroom
Physics Interactives by The Physics Classroom · Licensed to MarkScheme (site terms otherwise permit linking only)
3JCN PhysicsStart here · 59702 1.4 · IB A.1
Vector Subtraction
Subtract vectors graphically and read the resulting components
Why this one: A minus B is A plus the reversed B: multiplying by −1 flips a vector's direction.
Try this
1. Subtract two equal vectors and read the result.
2. Reverse the order of subtraction and compare the resulting components.
Look for A minus B equals A plus the reverse of B, so swapping the order reverses the result.
Open on 3JCN Physics
3JCN Physics Simulation by Thomas Nguyen · CC BY 4.0

More simulations21 more on this topic — core ones first

oPhysicsCore9702 1.4 · IB A.1
Vector Addition and Subtraction
Set magnitude/direction of two vectors with sliders; toggle to show sum or difference vector
Try this
1. Set both magnitudes equal and the directions 90° apart; show the sum.
2. Toggle to the difference vector without changing the sliders.
3. Slide one direction round until the sum is as short as it gets.
Look for The difference A − B is A plus the reversed B, so sum and difference are the two diagonals of the same parallelogram.
Open on oPhysics
Simulation by Tom Walsh, oPhysics.com — made with GeoGebra · Licensed to MarkScheme (site: free for non-profit educational use; applets made with GeoGebra)
oPhysicsCore9702 1.4 · IB A.1
Vector Addition and Subtraction Practice
Practice worksheet: draw the sum or difference of two given vectors, then check the answer
Try this
1. Draw the sum of the two given vectors tip to tail, then check the answer.
2. Draw the difference for the same pair and check it.
3. Generate a new pair and repeat.
Look for The sum closes the tip-to-tail chain; the difference joins the two tips when the vectors share a tail.
Open on oPhysics
Simulation by Tom Walsh, oPhysics.com — made with GeoGebra · Licensed to MarkScheme (site: free for non-profit educational use; applets made with GeoGebra)
oPhysicsCore9702 1.4 · 9702 2.1 · IB A.1
Relative Velocity: Boat Crossing a River
Set boat heading, boat speed and river speed; watch the resultant path across the river
Try this
1. Point the boat straight across with the river flowing and watch the path.
2. Angle the heading upstream until the path is straight across.
3. Set the river speed equal to the boat speed and try to cross directly.
Look for The boat’s path follows the vector sum of boat velocity and river velocity, so heading and path differ whenever the river flows.
Open on oPhysics
Simulation by Tom Walsh, oPhysics.com — made with GeoGebra · Licensed to MarkScheme (site: free for non-profit educational use; applets made with GeoGebra)
The Physics ClassroomCore9702 1.4 · IB A.1
Vector Addition
Drag up to three vectors onto the canvas, turn each arrowhead to set its direction, guess the resultant, then tap to draw it
Try this
1. Drag two vectors on and guess the direction of the resultant before drawing it.
2. Turn one arrowhead to reverse that vector and draw the resultant again.
3. Add a third vector and compare the resultant with your guess.
Look for The resultant runs from the tail of the first vector to the head of the last, whatever the arrowheads point.
Open on The Physics Classroom
Physics Interactives by The Physics Classroom · Licensed to MarkScheme (site terms otherwise permit linking only)
The Physics ClassroomCore9702 1.4 · IB A.1
Vector Walk in Two-Dimensions
Watch a person hike a two-dimensional path, then give the distance and the displacement (magnitude and direction); pass levels to progress
Try this
1. Add the lengths of each leg of the hike to get the distance.
2. Find the displacement magnitude and direction from start to finish.
3. Pass the level and try a hike with more legs.
Look for Distance is the sum of every leg of the hike; displacement is the single vector from start to finish.
Open on The Physics Classroom
Physics Interactives by The Physics Classroom · Licensed to MarkScheme (site terms otherwise permit linking only)
The Physics ClassroomCore9702 1.4 · 9702 2.1 · IB A.1
The Riverboat Simulator
Cross a river in a boat: set the boat speed, the river speed and the heading, then watch the resulting motion and read the crossing time
Try this
1. Set the river speed to zero and read the crossing time.
2. Add a river speed and compare the crossing time and the landing point.
3. Change the heading and watch where the boat lands.
Look for The river speed adds a velocity component along the bank that leaves the crossing time unchanged.
Open on The Physics Classroom
Physics Interactives by The Physics Classroom · Licensed to MarkScheme (site terms otherwise permit linking only)

Key formulas

Tap any symbol to reveal exactly what it means and its units.

KEY RELATION

$R = \sqrt{V_x^2 + V_y^2}$

Tap a symbol — great for exam definitions

KEY RELATION

Tap a symbol — great for exam definitions

Full topic notes

Formal explanation with the rigour you need for the exam.

Understanding Scalars

A scalar quantity is defined solely by its magnitude – essentially, its size or amount. Scalars have no associated direction. Think of how you measure your mass on a scale; it's just '70 kg', not '70 kg upwards'. These quantities simplify many everyday measurements, focusing only on 'how much'.

1.4 Scalars and vectors.
A scalar is a quantity which only has a magnitude.
E.g. speed, mass, time and distance.
A vector is a quantity with both a magnitude and direction.
Eg. velocity, acceleration, weight, and displacement.
If you want to know if a unit is a scalar or vector try putting a negative in front of it!.

Not clear?

Unveiling Vectors

In contrast, a vector quantity requires both a magnitude and a specific direction to be fully described. For instance, when you talk about your displacement, it's not just '5 meters' but '5 meters North'. Vectors are often represented visually as an arrow; the arrow's length corresponds to the vector's magnitude, and the arrowhead points in its direction.

Defined by magnitude and direction.
Represented by arrows (length = magnitude, arrowhead = direction).
Examples: displacement, velocity, force.
Examples: acceleration, momentum, electric field strength.

Not clear?

Adding Perpendicular Vectors

When two vectors act at right angles (perpendicular) to each other, combining them to find their resultant vector (the single vector equivalent to their combined effect) is straightforward. Imagine two forces, one pulling right and one pulling up. The overall effect, the resultant, can be found using fundamental geometry.

Resultant Magnitude (Pythagoras' Theorem): $R = \sqrt{V_x^2 + V_y^2}$

Resultant Direction (Trigonometry): $\tan \theta = \frac{\text{opposite}}{\text{adjacent}} = \frac{V_y}{V_x}$

Not clear?

Resolving Vectors

Sometimes, we need to do the reverse of adding: break down a single vector into two components that are at right angles to each other. This process is called vector resolution. It's incredibly useful when a force or velocity acts at an angle, and you need to see its individual effects along, say, horizontal and vertical axes.

If a vector $V$ makes an angle $\theta$ with the horizontal (x-axis): Component adjacent to the angle: $V_x = V \cos \theta$ Component opposite to the angle: $V_y = V \sin \theta$

Not clear?

Adding Non-Perpendicular Vectors

When vectors aren't at right angles, you can't use Pythagoras directly. One common and accurate method is to first resolve each vector into its perpendicular components (e.g., horizontal and vertical). Once you have all the horizontal components and all the vertical components, you can add them algebraically to get total horizontal ( $R_x$ ) and total vertical ( $R_y$ ) resultants. Then, combine these two total perpendicular components using Pythagoras' theorem and trigonometry to find the final resultant.

Alternatively, for a more visual approach, you can use scale diagrams and the head-to-tail method. Draw the first vector to scale, then draw the second vector (also to scale) starting from the head (tip) of the first. The resultant is the vector drawn from the tail (start) of the first to the head of the last.

EXAM TIP

For AS-Level Physics, the analytical method of resolving vectors into components is generally preferred over scale diagrams for accuracy, especially in Paper 2 calculations. Practice both, but master the component method!

Not clear?

Scalar Multiplication of Vectors

Multiplying a vector by a scalar changes its magnitude. If you multiply a vector by a positive scalar, its magnitude changes proportionally, but its direction remains the same. For example, $2\mathbf{v}$ is a vector twice as long as $\mathbf{v}$ but in the identical direction.

If you multiply a vector by a negative scalar, its magnitude also changes proportionally, but its direction is reversed. So, $-3\mathbf{v}$ would be a vector three times the length of $\mathbf{v}$ but pointing in precisely the opposite direction.

Not clear?

Worked examples

See the formulas applied — reveal one step at a time, like the exam.

reveal slowly — mark each step

EXAMPLE 1

A boat is travelling at 3.0 m/s due East, and simultaneously, a current pushes it 4.0 m/s due North. Calculate the magnitude and direction of the boat's resultant velocity.

1
Identify components: We have a horizontal velocity component $V_x = 3.0 \text{ m/s}$ (East) and a vertical velocity component $V_y = 4.0 \text{ m/s}$ (North). These are perpendicular.

EXAMPLE 2

Two forces, $F_1 = 60$ N and $F_2 = 40$ N, act on a small object P. $F_1$ acts horizontally to the right. $F_2$ acts at an angle of $60^\circ$ above the horizontal. Calculate the magnitude and direction of the resultant force acting on P.

1
Strategy: Resolve each force into horizontal (x) and vertical (y) components. Sum the components, then recombine them to find the resultant. Let's define 'right' and 'up' as positive directions.

How it all connects

The big idea sits in the middle — tap a linked idea to explore the link.

MAIN IDEAScalars and vectors

Tap a linked idea to see how it connects back to the main topic — that connection is what examiners reward.

Glossary

Key terms for this topic — skim now; the Check step will test them.

scalar quantity: A scalar quantity is defined solely by its magnitude – essentially, its size or amount.
resultant vector: When two vectors act at right angles (perpendicular) to each other, combining them to find their resultant vector (the single vector equivalent to their combined effect) is straightforward.
vector resolution: This process is called vector resolution. It's incredibly useful when a force or velocity acts at an angle, and you need to see its individual effects along, say, horizontal and vertical axes.
Resultant Magnitude (Pythagoras' Theorem):: Resultant Magnitude (Pythagoras' Theorem):
Resultant Direction (Trigonometry):: Resultant Direction (Trigonometry):
Component adjacent to the angle:: Component adjacent to the angle: $V_x = V \cos \theta$
Component opposite to the angle:: Component opposite to the angle: $V_y = V \sin \theta$

10Apply

Quick check

Write your answer first, then compare it with the model one — the gap is what you would have lost.

0 of 6 answered

11Understand

Teach it back

If you can explain it simply, you own it — gaps here are marks you’d lose.

Teach it back

Explain this topic as if teaching a friend. We name the gaps an examiner would still dock.

Your explanation — no notes

12Remember

Revision flashcards

Guess first, then flip — retrieval beats re-reading.

Your guess — write it before you look

1 / 15 · ← → space

Key takeaways

Review these before you close the topic — retrieval beats re-reading.

close the tab only after these stick

1.4 Scalars and vectors.
A scalar is a quantity which only has a magnitude.
E.g. speed, mass, time and distance.
A vector is a quantity with both a magnitude and direction.
Eg. velocity, acceleration, weight, and displacement.
If you want to know if a unit is a scalar or vector try putting a negative in front of it!.

Practice — then mark it

The whole point: a real Cambridge question, marked mark-by-mark.

Question 1 of 29702/23 · Q2(b)(iv)

PhysicsMay/June 2024

(b) (iv)

Determine the magnitude of the displacement of the object from its original position.

Do it on paper → mark itMARKED MARK-BY-MARK · B1 / M1 / A1 · OFFICIAL SCHEME

9702/23/M/J/24

Question 2 of 29702/22 · Q1(a)

PhysicsMay/June 2025

(a)

Table 1.1 lists some physical quantities. Identify with ticks (✔) which quantities are vectors and which are scalars.

Do it on paper → mark itMARKED MARK-BY-MARK · B1 / M1 / A1 · OFFICIAL SCHEME

9702/22/M/J/25

Extra simulations & links

PhET, GeoGebra and other curated tools — open in a new tab.

Frequently asked

What is the main difference between distance and displacement in physics?

Why is understanding vectors so important in physics?

Can a vector have zero magnitude but still possess a direction?

Checkpoint

One marked question is worth ten re-reads — close the loop before you move on.

Checkpoint

Reading it isn’t knowing it — prove it.

Before you move on: do 9702/23 · Q2(b)(iv) on paper, snap a photo, and get examiner-style feedback on exactly where you win and lose marks.

Attempt & get marked →Takes about a minute · you’ll land back on this lesson

← PREVIOUS1.3 Errors and Uncertainties in Paper 5 NEXT →2.1 Equations of motion

Discuss Scalars and vectors

Ask, share and discuss with other Physics students

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Revise Scalars and vectors for Cambridge 9702 (1.4)

Looking for free Physics 9702 notes on Scalars and vectors? This A-Level lesson covers official syllabus point 1.4 on Paper 1: Multiple Choice / Paper 2: AS Level Structured Questions. Use the visual steps, key terms, and quick checks below — then practise a real 9702 past-paper question with mark-scheme marking. Built for students searching "Scalars and vectors 9702 revision", "9702 1.4 notes", or a free alternative to paid note sites.

Full Physics 9702 course Mark a past paper →9702 subject hub

1.4Physics25 min readPREMIUM

Scalars and vectors

BY THE END, YOU CAN…

exam checklist — tick these off

1Distinguish between scalar and vector quantities, providing relevant examples.
2Add and resolve vectors using both graphical (scale diagrams) and analytical (component) methods.
3Understand the effect of multiplying a vector by a positive or negative scalar.

STUDY MODE

READING

TYPEFACE

Literata — a reading serif made for e-books; set like a textbook, sits well beside maths.

SIZE

AIRY SPACING

Wider letter and line spacing. The one change with solid evidence for readers who lose their place — worth a try whoever you are.

Pick whatever reads fastest for you — it differs from person to person, by up to a third.

EXPLAIN SIMPLER

The same notes in plain English — no jargon. Key points and exam tips stay.

DIAGRAM NOTES

A small sketch beside each block of notes.

In simple terms

A friendly intro before the formal notes — no formulas yet.

Scalars and vectors

Cambridge 9702 Paper 2 — Scalars and vectors (1.4). Senpai Corner diagram-backed pilot with premium structure and live visuals.

1
1.4 Scalars and vectors.
2
A scalar is a quantity which only has a magnitude.
3
E.g. speed, mass, time and distance.
4
A vector is a quantity with both a magnitude and direction.

What this topic covers

The official Cambridge syllabus points this lesson works through.

1.4.1
Understand the difference between scalar and vector quantities and give examples of scalar and vector quantities included in the syllabus
1.4.2
Add and subtract coplanar vectors
1.4.3
Represent a vector as two perpendicular components

Explore the concept

Use the live diagram, PhET or GeoGebra sim, and synced steps — play it, drag controls, or tap a step.

Figure PhET simulation1 / 4

Loading simulation…

Fetching from PhET…

Vector Addition

Resultant = tip-to-tail sum; components add independently.

If the embed does not load, use Open on PhET.

Simulation by PhET Interactive Simulations, University of Colorado Boulder · Licensed to MarkScheme (public licence CC BY-NC 4.0 since 2026-03-30)

Step-synced diagram — highlights what to look for in the simulation above.

Vectors have magnitude and direction

Vectors have magnitude and direction — show both on diagrams.

26 more simulations for this topic — run them in the Simulations section below

Simulations

Every simulation here runs the real model — try the steps on a card, then check what you see against the notes.

26 simulations · 5 to start with

Start herein this order — each one shows a different piece of the topic

oPhysicsStart here · 19702 1.4 · IB A.1 · IB A.2
Vector Addition
Drag the tips of two vectors; see their resultant by triangle or parallelogram method
Why this one: Triangle and parallelogram methods give the same resultant; watch it shrink as the vectors turn antiparallel.
Try this
1. Drag the two vector tips so they are perpendicular and read the resultant.
2. Switch between the triangle and parallelogram methods for the same pair.
3. Drag one tip until the two vectors point the same way.
Look for Both methods give the same resultant; it is longest when the vectors are parallel and shortest when antiparallel.
Open on oPhysics
Simulation by Tom Walsh, oPhysics.com — made with GeoGebra · Licensed to MarkScheme (site: free for non-profit educational use; applets made with GeoGebra)
The Physics ClassroomStart here · 29702 1.4 · 9702 2.1 · IB A.1
Vector Walk
Three short animated motions; work out the distance travelled and the overall displacement for each
Why this one: Add every step for distance, then draw start-to-finish for displacement: scalar versus vector in one walk.
Try this
1. Add up every step of the first motion to get the distance travelled.
2. Mark the start and finish and find the overall displacement.
3. Repeat for the other two motions and compare the two values each time.
Look for Distance counts every step of the path; displacement is only the straight-line change from start to finish, so it is never larger than distance.
Open on The Physics Classroom
Physics Interactives by The Physics Classroom · Licensed to MarkScheme (site terms otherwise permit linking only)
oPhysicsStart here · 39702 1.4 · IB A.1 · IB A.2
Vector Components
Enter a vector magnitude and direction; see its x and y components graphically and numerically
Why this one: Enter magnitude and angle, read |v| cos θ and |v| sin θ; past 90° the x-component goes negative.
Try this
1. Enter magnitude 10 at 30° and read both components.
2. Keep the magnitude and change the direction to 60°, then 90°.
3. Enter a direction between 90° and 180° and note the sign of the x component.
Look for The components are |v| cos θ and |v| sin θ, so they swap values between 30° and 60° and the x component turns negative past 90°.
Open on oPhysics
Simulation by Tom Walsh, oPhysics.com — made with GeoGebra · Licensed to MarkScheme (site: free for non-profit educational use; applets made with GeoGebra)
The Physics ClassroomStart here · 49702 1.4 · IB A.1
Name that Vector
Twelve challenges: add three of the 25 vectors shown on the grid by components and name the resultant
Why this one: Add x-components and y-components separately and name the resultant: the analytical method drilled.
Try this
1. Read the x- and y-components of each of the three vectors from the grid.
2. Add the x-components and the y-components separately.
3. Name the resultant from the two sums and check it.
Look for The components of the resultant are the sums of the x-components and of the y-components.
Open on The Physics Classroom
Physics Interactives by The Physics Classroom · Licensed to MarkScheme (site terms otherwise permit linking only)
3JCN PhysicsStart here · 59702 1.4 · IB A.1
Vector Subtraction
Subtract vectors graphically and read the resulting components
Why this one: A minus B is A plus the reversed B: multiplying by −1 flips a vector's direction.
Try this
1. Subtract two equal vectors and read the result.
2. Reverse the order of subtraction and compare the resulting components.
Look for A minus B equals A plus the reverse of B, so swapping the order reverses the result.
Open on 3JCN Physics
3JCN Physics Simulation by Thomas Nguyen · CC BY 4.0

More simulations21 more on this topic — core ones first

oPhysicsCore9702 1.4 · IB A.1
Vector Addition and Subtraction
Set magnitude/direction of two vectors with sliders; toggle to show sum or difference vector
Try this
1. Set both magnitudes equal and the directions 90° apart; show the sum.
2. Toggle to the difference vector without changing the sliders.
3. Slide one direction round until the sum is as short as it gets.
Look for The difference A − B is A plus the reversed B, so sum and difference are the two diagonals of the same parallelogram.
Open on oPhysics
Simulation by Tom Walsh, oPhysics.com — made with GeoGebra · Licensed to MarkScheme (site: free for non-profit educational use; applets made with GeoGebra)
oPhysicsCore9702 1.4 · IB A.1
Vector Addition and Subtraction Practice
Practice worksheet: draw the sum or difference of two given vectors, then check the answer
Try this
1. Draw the sum of the two given vectors tip to tail, then check the answer.
2. Draw the difference for the same pair and check it.
3. Generate a new pair and repeat.
Look for The sum closes the tip-to-tail chain; the difference joins the two tips when the vectors share a tail.
Open on oPhysics
Simulation by Tom Walsh, oPhysics.com — made with GeoGebra · Licensed to MarkScheme (site: free for non-profit educational use; applets made with GeoGebra)
oPhysicsCore9702 1.4 · 9702 2.1 · IB A.1
Relative Velocity: Boat Crossing a River
Set boat heading, boat speed and river speed; watch the resultant path across the river
Try this
1. Point the boat straight across with the river flowing and watch the path.
2. Angle the heading upstream until the path is straight across.
3. Set the river speed equal to the boat speed and try to cross directly.
Look for The boat’s path follows the vector sum of boat velocity and river velocity, so heading and path differ whenever the river flows.
Open on oPhysics
Simulation by Tom Walsh, oPhysics.com — made with GeoGebra · Licensed to MarkScheme (site: free for non-profit educational use; applets made with GeoGebra)
The Physics ClassroomCore9702 1.4 · IB A.1
Vector Addition
Drag up to three vectors onto the canvas, turn each arrowhead to set its direction, guess the resultant, then tap to draw it
Try this
1. Drag two vectors on and guess the direction of the resultant before drawing it.
2. Turn one arrowhead to reverse that vector and draw the resultant again.
3. Add a third vector and compare the resultant with your guess.
Look for The resultant runs from the tail of the first vector to the head of the last, whatever the arrowheads point.
Open on The Physics Classroom
Physics Interactives by The Physics Classroom · Licensed to MarkScheme (site terms otherwise permit linking only)
The Physics ClassroomCore9702 1.4 · IB A.1
Vector Walk in Two-Dimensions
Watch a person hike a two-dimensional path, then give the distance and the displacement (magnitude and direction); pass levels to progress
Try this
1. Add the lengths of each leg of the hike to get the distance.
2. Find the displacement magnitude and direction from start to finish.
3. Pass the level and try a hike with more legs.
Look for Distance is the sum of every leg of the hike; displacement is the single vector from start to finish.
Open on The Physics Classroom
Physics Interactives by The Physics Classroom · Licensed to MarkScheme (site terms otherwise permit linking only)
The Physics ClassroomCore9702 1.4 · 9702 2.1 · IB A.1
The Riverboat Simulator
Cross a river in a boat: set the boat speed, the river speed and the heading, then watch the resulting motion and read the crossing time
Try this
1. Set the river speed to zero and read the crossing time.
2. Add a river speed and compare the crossing time and the landing point.
3. Change the heading and watch where the boat lands.
Look for The river speed adds a velocity component along the bank that leaves the crossing time unchanged.
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Key formulas

Tap any symbol to reveal exactly what it means and its units.

KEY RELATION

$R = \sqrt{V_x^2 + V_y^2}$

Tap a symbol — great for exam definitions

KEY RELATION

Tap a symbol — great for exam definitions

Full topic notes

Formal explanation with the rigour you need for the exam.

Understanding Scalars

1.4 Scalars and vectors.
A scalar is a quantity which only has a magnitude.
E.g. speed, mass, time and distance.
A vector is a quantity with both a magnitude and direction.
Eg. velocity, acceleration, weight, and displacement.
If you want to know if a unit is a scalar or vector try putting a negative in front of it!.

Not clear?

Unveiling Vectors

Defined by magnitude and direction.
Represented by arrows (length = magnitude, arrowhead = direction).
Examples: displacement, velocity, force.
Examples: acceleration, momentum, electric field strength.

Not clear?

Adding Perpendicular Vectors

Resultant Magnitude (Pythagoras' Theorem): $R = \sqrt{V_x^2 + V_y^2}$

Resultant Direction (Trigonometry): $\tan \theta = \frac{\text{opposite}}{\text{adjacent}} = \frac{V_y}{V_x}$

Not clear?

Resolving Vectors

If a vector $V$ makes an angle $\theta$ with the horizontal (x-axis): Component adjacent to the angle: $V_x = V \cos \theta$ Component opposite to the angle: $V_y = V \sin \theta$

Not clear?

Adding Non-Perpendicular Vectors

EXAM TIP

Not clear?

Scalar Multiplication of Vectors

Not clear?

Worked examples

See the formulas applied — reveal one step at a time, like the exam.

reveal slowly — mark each step

EXAMPLE 1

A boat is travelling at 3.0 m/s due East, and simultaneously, a current pushes it 4.0 m/s due North. Calculate the magnitude and direction of the boat's resultant velocity.

1
Identify components: We have a horizontal velocity component $V_x = 3.0 \text{ m/s}$ (East) and a vertical velocity component $V_y = 4.0 \text{ m/s}$ (North). These are perpendicular.

EXAMPLE 2

1
Strategy: Resolve each force into horizontal (x) and vertical (y) components. Sum the components, then recombine them to find the resultant. Let's define 'right' and 'up' as positive directions.

How it all connects

The big idea sits in the middle — tap a linked idea to explore the link.

MAIN IDEAScalars and vectors

Tap a linked idea to see how it connects back to the main topic — that connection is what examiners reward.

Glossary

Key terms for this topic — skim now; the Check step will test them.

scalar quantity: A scalar quantity is defined solely by its magnitude – essentially, its size or amount.
resultant vector: When two vectors act at right angles (perpendicular) to each other, combining them to find their resultant vector (the single vector equivalent to their combined effect) is straightforward.
vector resolution: This process is called vector resolution. It's incredibly useful when a force or velocity acts at an angle, and you need to see its individual effects along, say, horizontal and vertical axes.
Resultant Magnitude (Pythagoras' Theorem):: Resultant Magnitude (Pythagoras' Theorem):
Resultant Direction (Trigonometry):: Resultant Direction (Trigonometry):
Component adjacent to the angle:: Component adjacent to the angle: $V_x = V \cos \theta$
Component opposite to the angle:: Component opposite to the angle: $V_y = V \sin \theta$

10Apply

Quick check

Write your answer first, then compare it with the model one — the gap is what you would have lost.

0 of 6 answered

11Understand

Teach it back

If you can explain it simply, you own it — gaps here are marks you’d lose.

Teach it back

Explain this topic as if teaching a friend. We name the gaps an examiner would still dock.

Your explanation — no notes

12Remember

Revision flashcards

Guess first, then flip — retrieval beats re-reading.

Your guess — write it before you look

1 / 15 · ← → space

Key takeaways

Review these before you close the topic — retrieval beats re-reading.

close the tab only after these stick

1.4 Scalars and vectors.
A scalar is a quantity which only has a magnitude.
E.g. speed, mass, time and distance.
A vector is a quantity with both a magnitude and direction.
Eg. velocity, acceleration, weight, and displacement.
If you want to know if a unit is a scalar or vector try putting a negative in front of it!.

Practice — then mark it

The whole point: a real Cambridge question, marked mark-by-mark.

Question 1 of 29702/23 · Q2(b)(iv)

PhysicsMay/June 2024

(b) (iv)

Determine the magnitude of the displacement of the object from its original position.

Do it on paper → mark itMARKED MARK-BY-MARK · B1 / M1 / A1 · OFFICIAL SCHEME

9702/23/M/J/24

Question 2 of 29702/22 · Q1(a)

PhysicsMay/June 2025

(a)

Table 1.1 lists some physical quantities. Identify with ticks (✔) which quantities are vectors and which are scalars.

Do it on paper → mark itMARKED MARK-BY-MARK · B1 / M1 / A1 · OFFICIAL SCHEME

9702/22/M/J/25

Extra simulations & links

PhET, GeoGebra and other curated tools — open in a new tab.

Frequently asked

What is the main difference between distance and displacement in physics?

Why is understanding vectors so important in physics?

Can a vector have zero magnitude but still possess a direction?

Checkpoint

One marked question is worth ten re-reads — close the loop before you move on.

Checkpoint

Reading it isn’t knowing it — prove it.

Before you move on: do 9702/23 · Q2(b)(iv) on paper, snap a photo, and get examiner-style feedback on exactly where you win and lose marks.

Attempt & get marked →Takes about a minute · you’ll land back on this lesson

← PREVIOUS1.3 Errors and Uncertainties in Paper 5 NEXT →2.1 Equations of motion

Discuss Scalars and vectors

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Revise Scalars and vectors for Cambridge 9702 (1.4)

In simple terms

What this topic covers

Explore the concept

Simulations

Vector Addition

Vector Walk

Vector Components

Name that Vector

Vector Subtraction

Vector Addition and Subtraction

Vector Addition and Subtraction Practice

Relative Velocity: Boat Crossing a River

Vector Addition

Vector Walk in Two-Dimensions

The Riverboat Simulator

Key formulas

Full topic notes

Understanding Scalars

Unveiling Vectors

Adding Perpendicular Vectors

Resolving Vectors

Adding Non-Perpendicular Vectors

Scalar Multiplication of Vectors

Worked examples

How it all connects

Glossary

Quick check

Teach it back

Revision flashcards

Key takeaways

Practice — then mark it

Extra simulations & links

Frequently asked

Checkpoint

Reading it isn’t knowing it — prove it.

Discuss Scalars and vectors

Revise Scalars and vectors for Cambridge 9702 (1.4)

In simple terms

What this topic covers

Explore the concept

Simulations

Vector Addition

Vector Walk

Vector Components

Name that Vector

Vector Subtraction

Vector Addition and Subtraction

Vector Addition and Subtraction Practice

Relative Velocity: Boat Crossing a River

Vector Addition

Vector Walk in Two-Dimensions

The Riverboat Simulator

Key formulas

Full topic notes

Understanding Scalars

Unveiling Vectors

Adding Perpendicular Vectors

Resolving Vectors

Adding Non-Perpendicular Vectors

Scalar Multiplication of Vectors

Worked examples

How it all connects

Glossary

Quick check

Teach it back

Revision flashcards

Key takeaways

Practice — then mark it

Extra simulations & links

Frequently asked

Checkpoint

Reading it isn’t knowing it — prove it.

Discuss Scalars and vectors