OCR AS Physics

Revision Notes

2.3.2 Combining Vectors

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 subtrating 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

Exam Tip

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

Combining Vectors with a Vector Triangle

  • Coplanar forces can be represented by the resultant of any two vector triangles
  • Forces are in equilibrium if an object is either
    • At rest
    • Moving at constant velocity
  • In equilibrium, coplanar forces are represented by closed vector triangles
    • The vectors, when joined together, form a closed path
  • The most common forces on objects are
    • Weight
    • Normal reaction force
    • Tension (from cords and strings)
    • Friction
  • The forces on a body in equilibrium are demonstrated below:

Vector triangle in equilibrium, downloadable AS & A Level Physics revision notes

Three forces on an object in equilibrium form a closed vector triangle

Worked Example

A weight hangs in equilibrium from a cable at point X. The tensions in the cables are T1 and T2 as shown.

WE - Forces in equilibrium question image 1, downloadable AS & A Level Physics revision notes

Which diagram correctly represents the forces acting at point X?

WE - Forces in equilibrium question image 2, downloadable AS & A Level Physics revision notes

Equilibrium Worked Example (3), downloadable AS & A Level Physics revision notes

Exam Tip

The diagrams in exam questions about this topic tend to be drawn to scale, so make sure you have a ruler handy!

Author: Ashika

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.
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