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?

• Both?column vectors?and?i, j notation?can be used for calculating?resultant vectors
• In both cases, a vector (force) 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 trigonometry but
  • treat?x?and?y?as?positive?(so |x| and |y| strictly speaking)
  • the angle found 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?column?vectors, a?resultant?force?of?zero would look like
  • 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’)

? ?(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