F = ma & Vector Notation (Edexcel IAL Maths)

Forces in 2D - Vector Notation

Spec point — 5.1

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

• If either (or both) components are negative then still use a diagram and $\mathrm{tan}θ^{\circ}=\frac{y}{x}$ but

  • treat x and y as positive (so |x| and |y| strictly speaking)
  • the angle found from $\mathrm{tan}θ^{\circ}=\frac{y}{x}$ 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

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 Two forces, (3i + 8j) N and (—5i — Ilj) N act on a particle.

(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 $(\xi+yj)N$

Answer:

Exam Hint

• 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.