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Learning outcomes

After studying this unit you should:

  • know some basic definitions and terminology associated with scalars and vectors and how to represent vectors in two dimensions;

  • understand how vectors can be represented in three (or more) dimensions and know both plane polar and Cartesian representations;

  • know ways to operate on and combine vectors.


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Introduction

This unit introduces the topic of vectors. The subject is developed without assuming you have come across it before, but the unit assumes that you have previously had a basic grounding in algebra and trigonometry, and how to use Cartesian coordinates for specifying a point in a plane.

This is an adapted extract from the Open University course Mathematical methods and models (MST209)
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All materials included in this unit are derived from content originated at the Open University.


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Acknowledgements

The content acknowledged below is Proprietary (see terms and conditions) and is used under licence.

All materials included in this unit are derived from content originated at the Open University.


Author(s): The Open University

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1 Modelling with Fourier series

The main teaching text of this unit is provided in the workbook below. The answers to the exercises that you'll find throughout the workbook are given in the answer book. You can access it by clicking on the link under the workbook.

Click 'View document' to open the workbook (PDF, 0.6 MB).

Learning outcomes

After studying this unit you should be able to:

  • understand how the wave and diffusion partial differential equations can be used to model certain systems;

  • determine appropriate simple boundary and initial conditions for such models;

  • find families of solutions for the wave equation, damped wave equation, diffusion equation and similar homogeneous linear second-order partial differential equations, subject to simple boundary conditions, using the meth
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Introduction

This unit shows how partial differential equations can be used to model phenomena such as waves and heat transfer. The prerequisite requirements to gain full advantage from this unit are an understanding of ordinary differential equations and basic familiarity with partial differential equations.

This unit is an adapted extract from the course Mathematical methods and models (MST209
Author(s): The Open University

Acknowledgements

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All materials included in this unit are derived from content originated at the Open University.


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1 First-order differential equations

The main teaching text of this unit is provided in the workbook below. The answers to the exercises that you'll find throughout the workbook are given in the answer book. You can access it by clicking on the link under the workbook.

Click 'View document' to open the workbook (PDF, 1.6 MB).

First-order differential equations

This unit introduces the topic of differential equations. The subject is developed without assuming that you have come across it before, but it is taken for granted that you have a basic grounding in calculus. In particular, you will need to have a good grasp of the basic rules for differentiation and integration.

This unit is an adapted extract from the course Mathematical methods and
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1.5 Exercises

Exercise 1

A vector a has magnitude |a| = 7 and direction θ = −70°. Calculate the component form of a, giving the components correct to two decimal places.

<
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1.4.3 Velocity

Another vector quantity which crops up frequently in applied mathematics is velocity. In everyday English, the words ‘speed’ and ‘velocity’ mean much the same as each other, but in scientific parlance there is a significant difference between them.

Velocity and speed

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1.4.2 Displacements and bearings

The displacement from a point P to a point Q is the change of position between the two points, as described by the displacement vector

If P and Q represent places on the ground, then it is natural to use a bearing to describe the direct
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1.4.1 Bearings

In the following subsections, we apply the vector ideas introduced so far to displacements and velocities. The examples will feature directions referred to points of the compass, known as bearings.

The direction of Leeds relative to Bristol can be described as ‘15° to the East of due North’, or N 15° E. This is an instance of a bearing. Directions on the ground are typically given like this, in terms of the directions North (N), South (S), East (E)
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1.3: Summing vectors given in geometric form

The following activity illustrates how the conversion processes outlined in the preceding sections may come in useful. If two vectors are given in geometric form, and their sum is sought in the same form, one approach is to convert each of the vectors into component form, add their corresponding components, and then convert the sum back to geometric form.

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1.2: Converting to geometric form

You have seen how any vector given in geometric form, in terms of magnitude and direction, can be written in component form. You will now see how conversion in the opposite sense may be achieved, starting from component form. In other words, given a vector a = a 1 i + a 2 j, what are its magnitude |a| and direction θ?

The first part of this question is dealt with using Pythagoras’ Theorem: the magnitude of a v
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1.1: Converting to component form

In some applications of vectors there is a need to move backwards and forwards between geometric form and component form; we deal here with how to achieve this.

To start with, we recall definitions of cosine and sine. If P is a point on the unit circle, and the line segment OP makes an angle θ measured anticlockwise from the positive x-axis, then cos θ is the x-coordinate of P and sin θ is the y-coordinate of P (
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Learning outcomes

On completion of this unit you should be able to:

  • convert a vector from geometric form (in terms of magnitude and direction) to component form;

  • convert a vector from component form to geometric form;

  • understand the use of bearings to describe direction;

  • understand the difference between velocity and speed;

  • find resultant displacements and velocities in geometric form, via the use of components.


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