Working with Vectors (AQA A Level Maths: Mechanics)

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Working with Vectors

Vectors are used throughout mechanics to describe forces and motion

 How are Vectors used in Mechanics?

  • Vector questions are often embedded in a Mechanics context
  • Vectors will most commonly represent forces, accelerations or velocities, they can also represent displacement
  • Newton’s Second Law bold F equals m bold ais essential
    • Remember that bold Fand bold a (force and acceleration) are vectors, while m (mass) is a scalar
  • When a particle is in equilibrium, the vector sum of the forces on it is equal to zero

What is vector notation?

  • There are two vector notations used in A level mathematics:
    • Column vectors: This is one number written above the other enclosed in brackets,

e.g. The (column) vector meaning 3 units in the positive horizontal (x) direction (i.e., right) and 2 units in the negative vertical (y ) direction (i.e., down) can be written as:

open parentheses table row 3 row cell negative 2 end cell end table close parentheses

   
    • i and j notation: i and j are unit vectors (they have magnitude 1) in the positive horizontal and positive vertical directions respectively

e.g. The vector (-4i + 3j) would mean 4 units in the negative horizontal (x) direction (i.e. left) and 3 units in the positive vertical (y) direction (i.e. up)

  • As they are vectors, i and j are displayed in bold in textbooks and online but in handwriting they would be underlined (i and j)

Calculating Resultant Vectors:

  • Both column vectors and i, j notation can be used for calculating resultant vectors
  • Adding vectors together gives the resultant vector
  • This is the same when adding force vectors; the resultant force is simply the force vectors added together
  • Forces in equilibrium have a resultant force equal to zero

1-2-1-working-with-vectors-diagram-1

Calculating Magnitude and Direction:

  • Pythagoras is used to find the magnitude of a vector.
  • Trigonometry is used to find the direction of a vector
  • You may be asked for the direction as a bearing. The unit vector j can be used to represent north and the unit vector i can be used to represent east.
    • In the case where k > 0
      • If a particle is moving north then its velocity will have a vector of kj
      • if a particle A is due south of another particle B then the displacement vector from B to A will have the form -kj
      • If a particle is moving east then its velocity will have a vector of ki
      • if a particle A is due west of another particle B then the displacement vector from B to A will have the form -ki
    • If the position vectors of two particles have the same j component, then the particle with the greater i component will be positioned due east of the other
    • If the position vector of a particle has equal i and j components then it is positioned due north - east of the origin
      • if a particle A is north – east of another particle B then the displacement vector from B to A will have equal i and j components

Magnitude Direction Diagram 1a

Resolving Vectors:

  • Resolving a vector means writing it in component form (as i and j components)
  • Given the magnitude and direction of a vector you can work out its components and vice versa

Worked example

1.2.1_WE_Working with Vectors_1

a)  Find the magnitude of the resultant force R acting on the particle.

     Leaving your answer as a simplified surd.

1-2-1-working-with-vectors-worked-solution-a

b)  Find the bearing of the resultant force R.

     Give your answer to the nearest degree.

1-2-1-working-with-vectors-worked-solution-b

A third force F3 = si – tj brings the particle into equilibrium.

c)   Find s and t and state the force for F3 in terms of i and j.

1-2-1-working-with-vectors-worked-solution-c

Examiner Tip

  • When working with vectors pay attention to accuracy, leaving magnitude in surd form or correct to 3.s.f.
  • In your exam you can’t write in bold so should underline your vector notation

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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.