## Exercise 1

A vector a has magnitude |a|Â =Â 7 and direction Î¸Â =Â âˆ’70Â°. Calculate the component form of a, giving the components correct to two decimal places.

Author(s): The Open University

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
Author(s): The Open University

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)
Author(s): The Open University

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.

Author(s): The Open University

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
Author(s): The Open University

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 x-axis, then cosÂ Î¸ is the x-coordinate of P and sinÂ Î¸ is the y-coordinate of P (
Author(s): The Open University

After studying this course, 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.

Author(s): The Open University

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

Course image: fdecomite in Flickr made available under Creative Commons Attribution 2.0 Licence.

Don't miss out:

If reading this text has inspired you to learn m
Author(s): The Open University

Hart, K., Kerslake, D., Brown, M., Ruddock, G., Kuchemann, D. and McCartney, M. (eds) (1981) Children's Understanding of Mathematics 11-16, London, John Murray.
McCague, W. (2003) 'A mathematical look at a medieval cathedral', Math Horizons, April, pp.11-15 and p.31. See also http://www.maa.org.
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Author(s): The Open University

The content acknowledged below is Proprietary (see and conditions made available under a Creative Commons Attribution-NonCommercial-ShareAlike 4.0 Licence) and used under licence.

Course image: rod
Author(s): The Open University

The doâ€“talkâ€“record triad (DTR) is a description of what is likely to take place in collaborative mathematics classrooms. It is concerned with observable events, and with the learner rather than the teacher, though many teaching insights flow from it. Although the order of the triad suggests that it should be followed in a particular sequence, this is not necessarily the case. Sometimes talking comes before doing or recording before talking. It also takes time for a learner to move
Author(s): The Open University

## 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 Author(s): The Open University

One of the characteristics of â€˜goodâ€™ information is that it should be balanced and present both sides of an argument or issue. This way the reader is left to weigh up the evidence and make a decision. In reality, we recognise that no information is truly objective.

This means that the onus is on you, the reader, to develop a critical awareness of the positions represented in what you read, and to take account of this when you interpret the information. In some cases, authors may
Author(s): The Open University

You can find a lot of information about the maths and statistics on the internet.

To find this information you might choose to use:

• search engines and subject gateways;

• books and electronic books;

• databases;

• journals;

• encyclopedias

• internet resources

Author(s): The Open University

In this course we will focus on the Anthropoidea, a suborder of primates that includes monkeys, apes and humans. We will concentrate our attention here primarily on monkeys. Colour vision, a large brain and intelligence are of great importance in the lives of anthropoids, enabling them to eat foods inaccessible to many other animals and to exploit social situations. In this course, we will be looking at characteristics of primates that differ, or are enhanced, in anthropoids and discussing th
Author(s): The Open University

Moderate NGOs, progressive businesses and government all have a stake in seeing roundtable partnerships come up with practical steps that can bring sustainability closer. One area that has attracted the attention of all these players is consumption. Directing or limiting consumption is politically difficult for even the NGOs to promote. Similarly, â€˜voluntary simplicityâ€™ of the sort lived at Findhorn eco-village (Author(s): The Open University

The idea underlying complementary currencies â€“ that there is a great well of social capital waiting to be drawn upon to make society more sustainable â€“ is an idea that is becoming quietly influential. â€˜Social capitalâ€™ is a term frequently used by those mainstream politicians and civil servants tasked with addressing the widening gap between rich and poor people within societies throughout the world. Indeed, investing in and enhancing social capital is now the starting point in
Author(s): The Open University

Governance is from the Greek words kybenan and kybernetes, meaning â€˜to steerâ€™ and â€˜pilotâ€™ or â€˜helmsmanâ€™. It is the process whereby â€˜an organization or a society steers itself, and the dynamics of communication and control are central to the processâ€™ (Rosenau and Durfee, 1995, p. 14). Of course, you could read these words as a pretty sound definition of government but that would be missing the point. Government describes a more rigid and narrower set
Author(s): The Open University

Human societies have to take urgent action to end their dependence on fossil fuels. They also have to prepare to adapt to the uncertainties inherent in global environmental changes, particularly climatic ones. We have to alter the whole path of our development and decision making in order to make our societies both environmentally adaptable and sustainable. This unit takes on the task of trying to chart some of the ways in which this might come about.

The context for these changes by g
Author(s): The Open University

Greenland snowfall differs depending on whether it falls in summer (when snow is comparatively warm and moist) or winter (when snow is cold and dry). These differences mean that as the snow is turned to ice, annual layers are formed that are in many ways similar to tree rings: thick annual layers mean high snowfall and thin annual layers low snowfall. The accumulation of snowfall on the summit of Greenland â€“ and most importantly what is trapped within the crystals as it turns to ice â€“ can
Author(s): The Open University