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2.3 Addition of vectors

We saw above that the vector 2v can be regarded as the vector v ‘followed by’ the vector v; we can also quite naturally describe this vector as being the ‘sum’, v + v, of the vector with itself.

Analogously, if p is the vector 2 cm E and q is the vector 3 cm NE, we can think of the ‘sum’ p + q of the vectors as follows. Starting from a given point, O say, draw the vector p; starting from its finishi
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2.2 Multiplication by a scalar

In the collection of vectors sketched in Section 2.1, although v is not equal to c, the vectors v and c are closely related: c is a vector in the same direction as v, but it is twice as long as v. Thus it is natural to write 2v for c, since we can think of a journey repre
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2.1 Definitions

In this section we introduce an alternative way of describing points in the plane 2 or in three-dimensional space 3;
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1.10 Further exercises

Exercise 12

Determine the equation of the line through each of the following pairs of points. Show that both equations can be written in the form ax + by = c, for some real numbers a
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1.9 Distance between points in three-dimensional Euclidean space

You saw in Section 1.5 that the distance between two points (x1, y1) and (x2, y2) in the plane is given by Author(s): The Open University

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1.8 Intersection of two planes

We saw earlier that two arbitrary lines in 2 may intersect, be parallel, or coincide. In an analogous way, two arbitrary planes in
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1.7 Planes in three-dimensional Euclidean space

We now look at the general form of the equation of a plane in 3.

Three planes whose equations are easy to find are those that contain a pair of axes. For example, the (x, y)-plane is the plane that contains the x-axis and the y-axis. Points which lie in this plane are prec
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1.5 Distance between two points in the plane

Next, we find the formula for the distance between two points P (x1, y1) and Q(x2, y2) in the plane. In the diagram below we have drawn P and Q in the first quadrant, but the formula we derive holds wherever the points are in the plane.


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1.4 Intersection of two lines

Two arbitrary lines in 2 may have a single point of intersection, may be parallel, or may coincide. The first two possibilities are illustrated below. Can we tell from the equations of the lines which of the three possibilities occurs?

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1.3 Parallel and perpendicular lines

We often wish to know whether two lines are parallel (that is, they never meet) or perpendicular (that is, they meet at right angles).

Two distinct lines, y = m1x + c1 and y = m2x + c2, are parallel if and only if they have the same gradient; that is, if and only if m1 = m2. For example, the lines y = −2x + 7 and
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1.2 Lines

The equation of any line in 2, except a line parallel to the y-axis, can be written in the form


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Learning outcomes

By the end of this unit you should be able to:

  • Section 1

  • recognise the equation of a line in the plane;

  • determine the point of intersection of two lines in the plane, if it exists;

  • recognise the one-one correspondence between the set of points in three-dimensional space and the set of ordered triples of real numbers;

  • recognise the equation of a plane in three dimensions.


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5.1 Leading the fight against slavery

Wilberforce’s name has been most famously associated with the issue of slavery. His success as a leader of the cause against slavery stemmed from his capacity to marshal a formidable range of argumentation, including secular as well as spiritual factors, and practical considerations as well as statements of principle. This section will examine extracts from two of Wilberforce’s writings on slavery, A Letter on the Abolition of the Slave Trade (1807) and An Appeal to the Religion
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Lesson 04 - One Minute Luxembourgish
In lesson 04 of One Minute Luxembourgish you will learn how to say that you don't understand something. Remember - even a few phrases of a language can help you make friends and enjoy travel more. Find out more about One Minute Languages at our website - http://www.oneminutelanguages.com. One Minute Luxembourgish is brought to you by the Radio Lingua Network and is ©Copyright 2008.Author(s): No creator set

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Season 3 – Lesson 08 – Coffee Break Spanish
Coming up in this lesson:

In episode 8, Alba and Mark discuss smoking laws in different countries. Language points covered include the future perfect tense, eg. “I will have stopped smoking”, the use of soler, and the phrase pillar a alguien desprevenido. José’s intermedio covers two phrases which you can use to describe how well you speak Spanish.

Please note that lesson 08 of Season 3 was originally known as lesson 308 of Coffee Break Spanish. We have ren
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1 Wave energy

The energy carried by ocean waves derives from a proportion of the wind energy transferred to the ocean surface by frictional drag. So, ultimately it stems from the proportion of incoming solar energy that drives air movement. Just how much energy is carried by a single wave depends on the wind speed and the area of ocean surface that it crosses; wave height, wavelength, and therefore wave energy, are functions of the distance or fetch over which the wind blows. Not surprisingly the ma
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3.2.2 The protection of intellectual property: the costs of TRIPS

Apart from the internal redistribution of income resulting from greater exposure to the world economy, the effects of one of the UR agreements in particular have achieved a certain notoriety because the agreement clearly imposes huge costs on farmers and consumers in developing countries, to the benefit of corporations in developed countries. This is the agreement on Trade-Related Aspects of Intellectual Property Rights (TRIPS), which strengthens international rules governing patents, tradema
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9.2 Lennox Castle Hospital

This course looks at the history of institutions in the twentieth century, starting with a case study of Lennox Castle Hospital. It tries to make sense of the history of Lennox Castle, and of institutional life in general, through testimony of those who experienced institutions as inmates and as nurses, as well as through Erving Goffman's model of the ‘total institution’. It examines the social bases of segregation, the professionalisation of staff in asylums and institutions, and campaig
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Volcanic Hazards in New Zealand
Brad Scott describes the different volcano types and potential threats in New Zealand. This includes active volcano footage thanks to Geoff Mackley.  (04:44)
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Earthquake Probabilities and Forecasting
Scientist, Matt Gerstenberger, uses statistical models to forecast aftershock probabilities of earthquakes.  He explains his work and why it is important.  (05:28)
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