Introduction to Vectors | Mathematics for GCSE/IGCSE - Year 11 PDF Download

Basic Vectors

What are vectors?

• A vector is a type of number that has both a size and a direction.
• We focus on two-dimensional vectors, although vectors can exist in any number of dimensions.

Representing vectors

• Vectors are represented as arrows, where the arrowhead shows the direction and the length of the arrow indicates the vector's magnitude (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 stack

• Note that the order of the letters is important! Vector stack  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 in Transformation Geometry

In transformation geometry, translations are represented using column vectors.

• 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

• Proficiency with vectors extends beyond transformation geometry and encompasses various applications in physics, engineering, and mathematics.
• On a grid, whether or not x and y axes are present, vectors can still be represented in the column vector form (x y).

Multiplying a vector by a scalar

• Scalars are numerical quantities that possess magnitude but lack direction, unlike vectors.
• Multiplying a vector by a positive scalar alters its magnitude while preserving its direction.
• In the representation of a vector as a column vector, each element within the column vector is multiplied by the scalar when scalar multiplication is performed.

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

Adding and subtracting vectors

• Adding two vectors is defined geometrically, like this:

• Subtracting one vector from another is equivalent to adding the negative of the second vector to the first vector.

a - b = a + (-b)

• When vectors are represented as column vectors, performing addition or subtraction involves simply adding or subtracting the corresponding x and y coordinates of the vectors.
• For example:

Magnitude of a Vector

What is a vector? 

• Vectors serve multiple purposes in mathematics. In mechanics, they represent quantities like velocity, acceleration, and forces. In geometry, such as in the IGCSE curriculum, vectors are applied to concepts like translation. 
• Familiarity with fundamental vector operations is essential, including determining magnitude or modulus, often represented in column vector form.

What is the magnitude or modulus of a vector?

• The magnitude of a vector varies depending on its application.
  • For velocity, magnitude corresponds to speed.
  • In the context of force, magnitude denotes the strength of the force, typically measured in Newtons.
• The terms "magnitude" and "modulus" are synonymous when referring to vectors.
• In geometry, magnitude or modulus represents the distance of the vector, always yielding a positive value.
• The vector's direction is insignificant when determining magnitude.
• Magnitude or modulus is denoted by vertical lines, such as |a| indicating the magnitude of vector a.

How do I find the magnitude or modulus of a vector

• Pythagoras’ Theorem!