Exercise 1
A vector a has magnitude aÂ =Â 7 and direction Î¸Â =Â âˆ’70Â°. Calculate the component form of a, giving the components correct to two decimal places.
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
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 â€˜15Â° to the East of due Northâ€™,
or NÂ 15Â°Â E. This is an instance of a
bearing. Directions on the ground are typically given like
this, in terms of the directions NorthÂ (N),
SouthÂ (S), EastÂ (E)
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.
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 aÂ =Â a _{1} iÂ +Â a _{2} j, what are its magnitude a and direction Î¸?
The first part of this question is dealt with using
Pythagorasâ€™ Theorem: the magnitude of a v
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 Î¸ measured anticlockwise
from the positive xaxis, then
cosÂ Î¸ is the xcoordinate of
P and sinÂ Î¸ is the
ycoordinate of P (
In this unit you will see first how to convert vectors from geometric form, in terms of a magnitude and direction, to component form, and then how conversion in the opposite sense is accomplished. The ability to convert between these different forms of a vector is useful in certain problems involving displacement and velocity, as shown in SectionÂ 2, in which you will also work with bearings.
This unit is an adapted extract from the Open University course
3 Work on your own mathematics
Two activities are given below. You are asked to work on them in turn and to record not only your working, but observations on what you notice about your emotions as you work through step by step.
Activity 3 Constrained numbers
W
Task 10 The MÃ¶bius band
Take a long thin strip of paper (preferably squared or graph paper) about 30 cm by 3 cm. Give one end a half twist and then tape it together. This is a MÃ¶bius band as shown in
Mathematical processes are different from content in that they overarch the subject and are not thought of as hierarchical. A list of processes could contain:

problemsolving (including investigating);

mathematical modelling;

reasoning;

communicating;

making connections (including applying mathematics); and

using tools.
Each of the six processes liste
1.1 Experiences of learning mathematics
You will come to this unit with many memories of mathematics, both as a teacher and a learner. It may help if you start by recalling memories of learning mathematics and making a record of them in your notebook.
When you work on a task, get into the habit of having your notebook to hand to record your thinking. Use the notebook in any way that helps you to think about the work you have done. Some people find it helpful to divide a page into two columns using the lefthand side to record
After studying this unit, you should:
understand some current issues in mathematics education, such as the relationship of mathematics content to mathematics processes.
understand a variety of approaches to the teaching of mathematics such as 'dotalkrecord'
be able to approach mathematical problems and tasks in a flexible way.
The following files accompany the exercise in Section 4.2
Clicking on 'View document' below opens an extract from Section 4.2 of the unit (PDF, 1.7 MB) which accompanies the audio clips, also below. Listen to each of them in turn with the extracted pages open (you may like to print them out). Work on the problems at the appropriate places â€“ you'll find the answers at the foot of this page.
After studying this unit you should:
be able to perform basic algebraic manipulation with complex numbers;
understand the geometric interpretation of complex numbers;
know methods of finding the nth roots of complex numbers and the solutions of simple polynomial equations.
This unit is an adapted extract from the Open University course Complex analysis (M337)
This unit is devoted solely to complex numbers.
In Section 1, we define complex numbers and show you how to manipulate them, stressing the similarities with the manipulation of real numbers.
Section 2 is devoted to the geometric representation of complex numbers. You will find that
Technorati reports that over 100 000 new â€˜blogsâ€™ are created each day. Because these online diaries offer instant publishing opportunities, you potentially have access to a wealth of knowledge from commentators and experts (if they blog) in a wide range of fields. Most internet searches will turn up results from blogs, but there are some blogspecific search engines such as: Blogdigger
Online bookshops and some of the major search engines offer â€˜Alertsâ€™ services. These work by allowing you to set up a profile once you have registered on their site, and when there are items meeting your criteria you receive an email. The good thing about alerts is that you donâ€™t have to do anything once you have set up your profile. The downside, particularly with alerts services from the search engines, is that given the extent to which internet traffic is on the increase whether new
1.5.6 Copyright – what you need to know
An original piece of work, whether it is text, music, pictures, sound recordings, web pages, etc., is protected by copyright law and may often have an accompanying symbol (Â©) and/or legal statement. In the UK it is the Copyright, Designs and Patents Act 1988 which regulates this.
In most circumstances, works protected by copyright can be used in whole or in part only with the permission of the owner. In some cases this permission results in a fee.
However, the UK legislation incl
Finding your paperwork or electronic files can be a problem. You may find that even if you do have some sort of filing system, your structure soon gets quite large with files in multiple locations, which can be hard to navigate. You may find yourself making arbitrary decisions about which folder to place a document in. It may make sense now but in the future, when you look where you think it should be, itâ€™s not there.
At times like this you may resort to the search command from the Wi
The date when information was produced or published can be an important aspect of quality. This is not quite as simple as saying that 'good' information has to be up to date.
Activity
Here is an example of a news item from an onl