Vector Addition (CIE IGCSE Additional Maths)

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Lucy

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Lucy

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Vector Addition

What is vector addition?

  • Adding vectors together lets us describes the movement between two points
  • To add or subtract vectors numerically simply add or subtract each of the corresponding components
  • In column vector notation just add the top, middle and bottom parts together
    • For example: open parentheses 2 over 1 close parentheses minus open parentheses 1 fourth close parentheses equals blank open parentheses fraction numerator 1 over denominator negative 3 end fraction close parentheses
  • In base vector notation add each of the i and j components together separately
    • For example: (2i + j) – (i + 4j) = (i – 3j)
       

Vector Addition Diagram 1b

  • Adding vectors creates a single vector which is called the resultant vector
    • The resultant vector will be the shortest route from the start of the first vector to the end of the second
  • Subtracting a vector is the same as adding a negative vector
  • Adding the vectors PQ and QP gives the zero vector, denoted by a bold zero 0 (0 in handwriting)

Vector Addition Diagram 1a, AS & A Level Maths revision notes

What are scalars and parallel vectors?

  • Two vectors are parallel if and only if one is a scalar multiple of the other
    • i.e both components of the vector have been multiplied by the same constant 
  • Multiplying a vector by a positive scalar changes the magnitude (size) but not its direction
  • Multiplying a vector by a negative scalar changes the magnitude and the direction would be reversed 

Vector Addition Diagram 2, AS & A Level Maths revision notes

How do I find the vector between two points?

  • If, relative to the origin O, the points A and B have position vectors
    • stack O A with rightwards arrow on top equals bold a
    • stack O B with rightwards arrow on top equals bold b

then

    • stack A B with rightwards arrow on top equals stack A O with rightwards arrow on top plus stack O B with rightwards arrow on top equals negative stack O A with rightwards arrow on top plus stack O B with rightwards arrow on top equals negative bold a plus bold b equals bold b minus bold a
  • Similarly, stack B A with rightwards arrow on top equals stack B O with rightwards arrow on top plus stack O A with rightwards arrow on top equals straight a minus straight b
  • This result is particularly useful when working with position vectors (as the 'journey' can always go via the origin)
    • but the result applies to any set of three vectors

Examiner Tip

  • Think of vectors like a journey from one place to another
    • You may have to take a detour eg. A to B might be A to O then O to B.
  • Diagrams can help, so if there isn’t one, draw one
    • If there are any, labelling parallel vectors will help

Worked example

Vector Addition Example Solution, AS & A Level Maths revision notes

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Lucy

Author: Lucy

Expertise: Head of STEM

Lucy has been a passionate Maths teacher for over 12 years, teaching maths across the UK and abroad helping to engage, interest and develop confidence in the subject at all levels. Working as a Head of Department and then Director of Maths, Lucy has advised schools and academy trusts in both Scotland and the East Midlands, where her role was to support and coach teachers to improve Maths teaching for all. Lucy has created revision content for a variety of domestic and international Exam Boards including Edexcel, AQA, OCR, CIE and IB.