Scalars & Vectors (AQA AS Physics)

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Katie M

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Katie M

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Scalars & Vectors

  • 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 them to produce the resultant vector
    • The resultant vector is sometimes known as the ‘net’ vector (eg. the net force)

  • There are two methods that can be used to add vectors
    • Calculation – if the vectors are perpendicular
    • Scale drawing – if the vectors are not perpendicular

Vector Calculation

  • Vector calculations will be limited to two vectors at right angles
  • This means the combined vectors produce a right-angled triangle and the magnitude (length) of the resultant vector is found using Pythagoras’ theorem

Magnitude of Vectors, downloadable AS & A Level Physics revision notes

The magnitude of the resultant vector is found by using Pythagoras’ Theorem

  • The direction of the resultant vector is found from the angle it makes with the horizontal or vertical
    • The question should imply which angle it is referring to (ie. Calculate the angle from the x-axis)

  • Calculating the angle of this resultant vector from the horizontal or vertical can be done using trigonometry
    • Either the sine, cosine or tangent formula can be used depending on which vector magnitudes are calculated

Direction of Vectors, downloadable AS & A Level Physics revision notes

The direction of vectors is found by using trigonometry

Scale Drawing

  • When two vectors are not at right angles, the resultant vector can be calculated using a scale drawing
    • Step 1: Link the vectors head-to-tail if they aren’t already
    • Step 2: Draw the resultant vector using the triangle or parallelogram method
    • Step 3: Measure the length of the resultant vector using a ruler
    • Step 4: Measure the angle of the resultant vector (from North if it is a bearing) using a protractor

Scale Drawing, downloadable AS & A Level Physics revision notes

A scale drawing of two vector additions. The magnitude of resultant vector R is found using a rule and its direction is found using a protractor

  • Note that with scale drawings, a scale may be given for the diagram such as 1 cm = 1 km since only limited lengths can be measured using a ruler
  • The final answer is always converted back to the units needed in the diagram
    • Eg. For a scale of 1 cm = 2 km, a resultant vector with a length of 5 cm measured on your ruler is actually 10 km in the scenario

  • There are two methods that can be used to combine vectors: the triangle method and the parallelogram 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 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

Vector Addition

Vector Addition, downloadable AS & A Level Physics revision notes

Vector Subtraction

Vector Subtraction, downloadable AS & A Level Physics revision notes

Worked example

A hiker walks a distance of 6 km due east and 10 km due north. Calculate the magnitude of their displacement and its direction from the horizontal

4.1.1 Vector Addition Worked Example

Examiner Tip

Pythagoras' Theorem and trigonometry are consistently used in vector addition, so make sure you're fully confident with the maths here!

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Katie M

Author: Katie M

Expertise: Physics

Katie has always been passionate about the sciences, and completed a degree in Astrophysics at Sheffield University. She decided that she wanted to inspire other young people, so moved to Bristol to complete a PGCE in Secondary Science. She particularly loves creating fun and absorbing materials to help students achieve their exam potential.