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Scalars & Vectors (CIE AS Physics)

Revision Note

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What are scalar & vector quantities?

  • All quantities can be one of two types:
    • a scalar
    • a vector

Scalars

  • Scalars are quantities that have magnitude but not direction
    • For example, mass is a scalar quantity because it has magnitude but no direction 

Vectors

  • Vectors are quantities that have both magnitude and direction
    • For example, weight is a vector quantity because it is a force and has both magnitude and direction

Distance and displacement

  • Distance is a measure of how far an object has travelled, regardless of direction
    • Distance is the total length of the path taken
    • Distance, therefore, has a magnitude but no direction
    • So, distance is a scalar quantity
  • Displacement is a measure of how far it is between two points in space, including the direction
    • Displacement is the length and direction of a straight line drawn from the starting point to the finishing point
    • Displacement, therefore, has a magnitude and a direction
    • So, displacement is a vector quantity

What is the difference between distance and displacement?

Displacement v distance

Displacement is a vector quantity while distance is a scalar quantity

  • When a student travels to school, there will probably be a difference in the distance they travel and their displacement
    • The overall distance they travel includes the total lengths of all the roads, including any twists and turns
    • The overall displacement of the student would be a straight line between their home and school, regardless of any obstacles, such as buildings, lakes or motorways, along the way

Speed and velocity

  • Speed is a measure of the distance travelled by an object per unit time, regardless of the direction
    • The speed of an object describes how fast it is moving, but not the direction it is travelling in
    • Speed, therefore, has magnitude but no direction
    • So, speed is a scalar quantity
  • Velocity is a measure of the displacement of an object per unit time, including the direction
    • The velocity of an object describes how fast it is moving and which direction it is travelling in
    • An object can have a constant speed but a changing velocity if the object is changing direction
    • Velocity, therefore, has magnitude and direction
    • So, velocity is a vector quantity

Examples of scalars & vectors

  • The table below lists some common examples of scalar and vector quantities

Table of scalars and vectors

Scalars Vectors
distance displacement
speed velocity
mass acceleration
time force
energy momentum
volume  
density  
pressure  
electric charge  
temperature  

Combining vectors

  • Vectors are represented by an arrow
    • The arrowhead indicates the direction of the vector
    • The length of the arrow represents the magnitude

  • Vectors can be combined by adding or subtracting them from each other
  • There are two methods that can be used to combine vectors: the triangle method and the parallelogram method

Triangle Method

  • To combine vectors using the triangle method:
    • Step 1: link the vectors head-to-tail
    • Step 2: the resultant vector is formed by connecting the tail of the first vector to the head of the second vector
    • To subtract vectors, change the direction of the vector from positive to negative and add them in the same way

Triangle method for adding and subtracting vectors

Introduction to Vectors | CIE IGCSE Maths: Extended Revision Notes 2025

The triangle method links vectors tip to tail to find the resultant vector

Parallelogram method

  • To combine vectors using the parallelogram method:
    • Step 1: link the vectors tail-to-tail
    • Step 2: complete the resulting parallelogram
    • Step 3: the resultant vector is the diagonal of the parallelogram

Parallelogram method for adding and subtracting vectors

Vector Subtraction 2, downloadable IB Physics revision notes

The parallelogram method links vectors tail to tail to find the resultant vector 

  • When two or more vectors are added together (or one is subtracted from the other), a single vector is formed, known as the resultant vector
  • The magnitude of the resultant vector can be found using Pythagoras' theorem or trigonometry

Condition for equilibrium

  • Coplanar forces can be represented by vector triangles
  • In equilibrium, these are closed vector triangles.
    • The vectors, when joined together, form a closed path

Forces in equilibrium

Vector Equilibrium, downloadable AS & A Level Physics revision notes

If three forces acting on an object are in equilibrium; they form a closed triangle

 

Worked example

A swimmer is crossing a river by swimming due north at 2.7 m s−1. The current flows east at 5.1 m s−1.

Determine the resultant velocity of the swimmer's motion.

Answer:

Step 1: Sketch a vector diagram of the scenario

combining-vectors-we

Step 2: List the known quantities

  • Velocity 1, v subscript 1 space equals space 2.7 space straight m space straight s to the power of negative 1 end exponent space straight N
  • Velocity 2, v subscript 2 space equals space 5.1 space straight m space straight s to the power of negative 1 end exponent space straight E

Step 3: Calculate the magnitude of the resultant vector, v

  • Using Pythagoras

v squared space equals space v subscript 1 squared space plus space v subscript 2 squared

v space equals space square root of v subscript 1 squared space plus space v subscript 2 squared end root

v space equals space square root of 2.7 squared space plus space 5.1 squared end root

v space equals space 5.8 space straight m space straight s to the power of negative 1 end exponent

Step 3: Calculate the direction of the resultant vector

  • Using trigonometry

tan space theta space equals space opposite over adjacent space equals space v subscript 1 over v subscript 2

theta space equals space tan to the power of negative 1 end exponent open parentheses fraction numerator 2.7 over denominator 5.1 end fraction close parentheses

theta space equals space 28 degree space open parentheses 2 space straight s. straight f. close parentheses

Resolving vectors

  • Two vectors can be represented by a single resultant vector that has the same effect
  • A single resultant vector can be resolved and represented by two vectors, which in combination have the same effect as the original one
  • When a single resultant vector is broken down into its parts, those parts are called components
  • For example, a force vector of magnitude F and an angle of θ to the horizontal is shown below

Resultant vector diagram

Representing Vectors, downloadable AS & A Level Physics revision notes

A resultant vector, F

  • It is possible to resolve this vector into its horizontal and vertical components using trigonometry

Horizontal and vertical vector components

Resolving Vectors, downloadable AS & A Level Physics revision notes

Horizontal and vertical components of F

 

  • For the horizontal component, Fx = Fcosθ
  • For the vertical component, Fy = Fsinθ

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Ashika

Author: Ashika

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Ashika graduated with a first-class Physics degree from Manchester University and, having worked as a software engineer, focused on Physics education, creating engaging content to help students across all levels. Now an experienced GCSE and A Level Physics and Maths tutor, Ashika helps to grow and improve our Physics resources.