Introduction to Vectors (Edexcel GCSE Maths)

Revision Note

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Naomi C

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Naomi C

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Maths

Basic Vectors

What are vectors?

  • A vector is a type of number that has both a size and a direction
  • Here we only deal with two-dimensional vectors, although it is possible to have vectors with any number of dimensions

Representing vectors

  • Vectors are represented as arrows, with the arrowhead indicating the direction of the vector, and the length of the arrow indicating the vector’s magnitude (ie its size):

Vector magnitude direction, IGCSE & GCSE Maths revision notes 

  • In print vectors are usually represented by bold letters (as with vector a in the diagram above), although in handwritten workings underlined letters are normally used, a.
  • Another way to indicate a vector is to write its starting and ending points with an arrow symbol over the top such as stack A B with rightwards arrow on top

Vector start end point, IGCSE & GCSE Maths revision notes

 

  • Note that the order of the letters is important! Vector stack B A with rightwards arrow on top in the above diagram would point in the opposite direction (ie with its ‘tail’ at point B, and the arrowhead at point A).

Vectors and transformation geometry

In transformation geometry, translations are indicated in the form of a column vector:

Translation Vector x,y, IGCSE & GCSE Maths revision notes

 

  • In the following diagram, Shape A has been translated six squares to the right and 3 squares up to create Shape B
  • This transformation is indicated by the translation vector open parentheses table row 6 row 3 end table close parentheses: Matrix Xlation Vector, IGCSE & GCSE Maths revision notes
  • Note: ‘Vector’ is a word from Latin that means ‘carrier’
  • In this case, the vector ‘carries’ shape A to shape B, so that meaning makes perfect sense!

Vectors on a grid

  • You also need to be able to work with vectors on their own, outside of the transformation geometry context
  • When vectors are drawn on a grid (with or without x and y axes), the vectors can be represented in the same (x y) column vector form as above

Vectors on grid, IGCSE & GCSE Maths revision notes

 bold a equals open parentheses table row 3 row 4 end table close parentheses comma space space bold b equals open parentheses table row 2 row cell negative 4 end cell end table close parentheses comma space space bold c equals open parentheses table row 2 row 0 end table close parentheses 

Multiplying a vector by a scalar

  • A scalar is a number with a magnitude but no direction – ie the regular numbers you are used to using
  • When a vector is multiplied by a positive scalar, the magnitude of the vector changes, but its direction stays the same
  • If the vector is represented as a column vector, then each of the numbers in the column vector gets multiplied by the scalar

Vector mult by scalar, IGCSE & GCSE Maths revision notes 

bold a equals open parentheses table row 4 row cell negative 2 end cell end table close parentheses comma space space space space space 2 bold a equals open parentheses table row cell 2 cross times 4 end cell row cell 2 cross times negative 2 end cell end table close parentheses equals open parentheses table row 8 row cell negative 4 end cell end table close parentheses comma space space space space 1 half bold a equals open parentheses table row cell 0.5 cross times 4 end cell row cell 0.5 cross times negative 2 end cell end table close parentheses equals open parentheses table row 2 row cell negative 1 end cell end table close parentheses

  • Note that multiplying by a negative scalar also changes the direction of the vector: 

Vector mult by neg scalar, IGCSE & GCSE Maths revision notes

bold a equals open parentheses table row 4 row cell negative 2 end cell end table close parentheses comma space space space space space minus bold a equals open parentheses table row cell negative 4 end cell row 2 end table close parentheses comma space space space space minus 2 bold a equals open parentheses table row cell negative 8 end cell row 4 end table close parentheses

  • Note in particular that vector -a is the the same size as vector a, but points in the opposite direction!

Adding and subtracting vectors

  • Adding two vectors is defined geometrically, like this:

Vector addition no grid, IGCSE & GCSE Maths revision notes 

  • Subtracting one vector from another is thought of as adding a negative vector Vector subtraction no grid, IGCSE & GCSE Maths revision notes

a – b = a + (-b)

  • When vectors are represented as column vectors, adding or subtracting is simply a matter of adding or subtracting the vectors’ x and y coordinates
  • For example:

Vector add and subtr grid, IGCSE & GCSE Maths revision notes 

bold a equals open parentheses table row 2 row cell negative 4 end cell end table close parentheses comma space space space space space bold b equals open parentheses table row 3 row 2 end table close parentheses space space space
bold a plus bold b equals open parentheses table row cell 2 plus 3 end cell row cell negative 4 plus 2 end cell end table close parentheses equals open parentheses table row 5 row cell negative 2 end cell end table close parentheses
bold a bold minus bold b equals open parentheses table row cell 2 minus 3 end cell row cell negative 4 minus 2 end cell end table close parentheses equals open parentheses table row cell negative 1 end cell row cell negative 6 end cell end table close parentheses

Worked example

The points A, B and C are shown on the following coordinate grid.Question points on grid, IGCSE & GCSE Maths revision notes

(a)

Write the vectors stack A B with rightwards arrow on top comma space stack A C with rightwards arrow on top and stack C B with rightwards arrow on top as column vectors.

Start by drawing the three vectors onto the grid.

Question points with vectors, IGCSE & GCSE Maths revision notes

From A to B, it is 6 to the right and 2 up.

  

From A to C, it is 7 to the right and 6 down.

  

From C to B, it is 1 to the left and 8 up.

  

(b)

Using the column vectors from (a), confirm that stack A B with rightwards arrow on top minus stack A C with rightwards arrow on top equals stack C B with rightwards arrow on top.

Just perform the subtraction on the column vectors.

  stack A B with rightwards arrow on top minus stack A C with rightwards arrow on top equals open parentheses table row 6 row 2 end table close parentheses minus open parentheses table row 7 row cell negative 6 end cell end table close parentheses equals open parentheses table row cell 6 minus 7 end cell row cell 2 minus open parentheses negative 6 close parentheses end cell end table close parentheses equals open parentheses table row cell negative 1 end cell row 8 end table close parentheses equals stack C B with rightwards arrow on top

stack bold italic A bold italic B with bold rightwards arrow on top bold minus stack bold italic A bold italic C with bold rightwards arrow on top bold equals stack bold italic C bold italic B with bold rightwards arrow on top

Length of a Vector

What is a vector?

Vec Mod Notes fig1, downloadable IGCSE & GCSE Maths revision notes

  • Vectors have various uses in mathematics
    • In mechanics vectors represent velocity, acceleration and forces
    • At GCSE vectors are used in geometry – eg. translation
    • Ensure you are familiar with the Revision Notes Vectors – Basics

  • These notes look at finding the length (also referred to as the magnitude, or modulus), of a vector
    • Vectors are given in column vector form
    • Vectors have length (magnitude) and direction

What is the length of a vector?

Vec Mod Notes fig2, downloadable IGCSE & GCSE Maths revision notes

  • This depends on the use of the vector
    • For velocity, magnitude would be speed
    • For a force, magnitude would be the strength of the force (in Newtons)

  • The words length, magnitude and modulus mean the same thing with vectors
  • In geometry magnitude and modulus mean the length or distance of the vector
    • This is always a positive value
    • The direction of the vector is irrelevant

  • Magnitude or modulus is indicated by vertical lines
    • |a| would mean the magnitude of vector a
    • You do not need to be use or understand this notation for your exam

How do I find the length of a vector

  • Pythagoras’ Theorem!

 Vec Mod Notes fig3, downloadable IGCSE & GCSE Maths revision notes

Exam Tip

  • Sketch a vector to help, it does not have to be to scale, then you can use this to form a right-angled triangle.

Worked example

Two points, P and Q, are plotted on a grid. Given that stack P Q with rightwards arrow on top equals open parentheses table row 8 row cell negative 6 end cell end table close parentheses find the length of the line segment that joins P and Q.

You can form a right angled triangle by starting at P and then going 8 to the right and 6 down to get to Q.

edexcel-vector-length-we-diagram

The length between P and Q is the hypotenuse of this triangle so you can use Pythagoras' theorem.

table row cell x squared end cell equals cell 8 squared plus 6 squared end cell row cell x squared end cell equals cell 64 plus 36 end cell row cell x squared end cell equals 100 row x equals cell square root of 100 end cell row x equals 10 end table

Length is 10 units

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Naomi C

Author: Naomi C

Naomi graduated from Durham University in 2007 with a Masters degree in Civil Engineering. She has taught Mathematics in the UK, Malaysia and Switzerland covering GCSE, IGCSE, A-Level and IB. She particularly enjoys applying Mathematics to real life and endeavours to bring creativity to the content she creates.