Introduction to Vectors (Cambridge O Level Maths)

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)

• 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 

• Note that the order of the letters is important! Vector  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:

• 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 :

• 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 

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

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

• 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:

• Subtracting one vector from another is thought of as adding a negative vector

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:

What is a vector?

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

• Vectors have magnitude and direction
  • These notes look at finding the magnitude, or modulus, of a vector
  • Vectors are given in column vector form

What is the magnitude or modulus of a vector?

• 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 magnitude and modulus mean the same thing with vectors
• In geometry magnitude and modulus mean the 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

How do I find the magnitude or modulus of a vector

• Pythagoras’ Theorem!