Forces in 2D - Vector Notation (Edexcel International A Level Maths: Mechanics 1)

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Forces in 2D - Vector Notation

How are forces and vectors linked?

  • Forces are vectors – they have magnitude and direction.
  • The magnitude of a force is measured in Newtons and the direction of a force is an angle, usually measured in degrees, anti-clockwise from the horizontal
  • There are other ways to talk about direction – in particular Cartesian coordinates(x, y) are used to describe the position of a point in two-dimensional space (plane), relative to a fixed origin O.
  • You may see questions start with a sentence along the lines of “Relative to the frame of reference Oxy …” – this means the two-dimensional space (plane) and distances within it will be based on horizontal (x) and vertical (y) components relative to the origin O.

What is vector notation?

  • Forces in 2D should be given in the form pi +qj where i and j are unit vectors
  • Both column vectors and i, j notation can be used for calculating resultant vectors
    • Column vectors can be useful when calculating with vectors as mistakes are less common, but you must remember to change the answer back to i, j notation form
  • A force written as a vector is written with its two components separated so the magnitude and direction of the vector are not directly known

What are the notations for magnitude and direction?

  • The magnitude of the force F N would be denoted by |F| N  or F N  – notice the use of bold and italics in particular
  • Direction is an angle, usually measured in degrees anti-clockwise from the horizontal, θ° is usually used

How do I find the magnitude and direction of a force from its components?

  • For a force F = xi + yj N to find
    • its magnitude, |F| N or F N, use Pythagoras’ theorem
    • its direction (as an angle), use a diagram and trigonometry

    3-1-4-fig1-finding-the-magnitude-and-direction

  • If either (or both) components are negative then still use a diagram and tan space theta degree space equals space y over x but
    • treat x and y as positive (so |x| and |y| strictly speaking)
    • the angle found from tan space theta degree space equals space y over x may need adjusting depending on where, and which way, the direction is being measured from
      e.g. Find the direction at which the force F = (-8i - 6j) N, giving your answer as an angle measured in degrees anti-clockwise from the positive horizontal direction

    3-1-4-fig2-negative-components

What does equilibrium with vectors mean?

  • In two dimensions a particle is in equilibrium if the resultant force acting on it in both directions is zero
    • For vectors in i-j notation, a resultant force of zero would look like (0i + 0j) N 
    • Both forms may be written as 0 N (called ‘the zero vector’)

Worked example

3.1.4_WE_Forces in 2D – Vector Notation_1

   (i)     Find the resultant force.

   (ii)    Find the magnitude of the resultant force and its direction as an angle measured
           anti-clockwise from the i-direction.

   (iii)   A third force is applied to the particle such that it is brought into equilibrium.
           Find the third force, giving your answer in the form left parenthesis x bold i plus y bold j right parenthesis space straight N

3-1-4-fig3-we-solution

Examiner Tip

  • Ideally you should stick to the i-,j- vector notation used in a question but if you prefer to use column vectors within calculations you can use a mixture within the same question.  Just be careful your final answer is in the correct format.
  • Draw diagrams – including ‘mini’-diagrams of individual forces/vectors – this can help in understanding a problem and being accurate.

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Amber

Author: Amber

Expertise: Maths

Amber gained a first class degree in Mathematics & Meteorology from the University of Reading before training to become a teacher. She is passionate about teaching, having spent 8 years teaching GCSE and A Level Mathematics both in the UK and internationally. Amber loves creating bright and informative resources to help students reach their potential.