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Mathematics: Vectors & Matrices

Mathematics support for students at the University of Suffolk.

Vectors and Matrices

Here you will find videos and worksheets on the topics of vectors and matrices.

  1. Vectors Introduction - What is a vector; magnitude of a vector
  2. Angles Between Vectors - The scalar product
  3. Vector Equation of a Line
  4. Vector Worksheet - A review worksheet, with solutions, on Vectors
  5. Matrices Introduction - Dimensions of a matrix; transpose of a matrix
  6. Matrix Arithmetic - Addition, subtraction and multiplication of matrices
  7. Inverses of Matrices - Determinants; inverses of matrices
  8. Vector and Matrices Worksheet - A review worksheet, with solutions, on Vectors and Matrices.

Vectors Introduction

Vectors

What is a vector?

This 10 minute youtube video, by 3Blue1Brown, covers:

  • what is a vector
  • vector addition
  • vector scaling

Vector notation

Vectors in space can be described using 2-d and 3-d column vectors.

The vector a is a 3-d vector. It represents a vector that goes 4 units in the positive x-direction, 3 units in the positive y-direction and 2 units in negative z direction.

Unit vectors can be defined in the x, y and z direction. The vector i is a vector that is 1 unit in the positive x direction only. We can describe vectors in terms of unit vectors. The vector above can also be written = 4i + 3j - 2z.

Magnitude of a Vector

The magnitude of a vector is the length of the vector. It can be found by making use of the Theorem of Pythagoras. Pythagoras in 3-d is very similar to Pythagoras in 2-d (which you would have done at school). The next 10 minute youtube video from ExamSolutions covers how to calculate the magnitude of a vector in 3-d space.

Angle Between Vectors

Angle Between Vectors

The 11 minute youtube video from ExamSolutions covers how to find the angle between two vectors. It covers

  • What angle are we talking about?
  • The formula for finding the cosine of the angle
  • The scalar (or dot) product
  • Examples of finding angles

Vector Equation of a Line

The Vector Equation of a Line

We may be familiar with the Cartesian equation of a straight line from school - y = mx + c. Straight lines can also be defined by a vector equation. For a vector equation of a line we need a position vector on the line (a vector which takes us from the origin to a point on the line) and a direction vector for the line (a vector which runs parallel to the straight line).

The two youtube videos from ExamSolutions will take you through how vector equations define a line (first video 9 minutes) and an example of finding the vector equation of a line from two points on a line (second video 8 minutes).

Matrices Introduction

Matrices

A matrix shows an array of values shown in rows and columns.

This 6 minute youtube video, from Mathsispower4u, demonstrates how to find the dimension of a matrix.

 

The next 4 minute youtube video from Khan Academy looks at matrix transposes. This is a simple operation whereby rows and columns of a matrix are switched.

Matrix Arithmetic

Matrix Arithmetic

The operations of addition, subtraction and multiplication can be performed on matrices.

The 6 minute youtube video from Khan Academy demonstrates how to complete addition and subtraction questions with matrices.

 

Multiplication of matrices is a more complicated process. In order to perform matrix multiplication the dimensions of each matrix need to match in a particular manner. If matrix A has dimensions p x q (p rows, q columns) then matrix B would need to have dimensions q x r (q rows, r columns) if we want to work out the matrix AB. The resultant matrix will have dimensions p x r. The order of multiplication is important, AB does not necessary equal BA.

The next two youtube Khan Academy videos (12 minutes) demonstrate the procedure to multiply two matrices together.

Inverses of Matrices

Inverse of a Matrix

Determinants

For a square matrix (number of rows = number of columns) a numerical value called the determinant. The larger the matrix the more difficult it is to calculate the determinant.

The youtube video from Mathispower4u (8 minutes) goes through the method to calculate the determinant of a 2x2 and 3x3 matrix.

 

If the determinant of a matrix is 0 then the matrix is called singular. Otherwise the matrix is known as non-singular.

For non-singular matrices an inverse can be found. The 3 minute youtube video from Khan Academy shows how to find the inverse of a 2x2 matrix.