4.6 Hyperbola (e > 1)

A hyperbola is the set of points P in the plane whose distances from a fixed point F are e times their distances from a fixed line d, where e > 1. We obtain a hyperbola in standard form if

1. the focus F lies on the x-axis, and has coordinates (ae, 0), where a > 0;

2. the directrix d is the line with equation x = a/e.

Author(s): The Open University

4.2 Circles

Recall that a circle in ² is the set of points (x, y) that lie at a fixed distance, called the radius, from a fixed point, called the centre of the circle. We can use the techniques of coordinate geometry to find the equation of a circle with a given centre and radius.
Author(s): The Open University

3.1 Definition, properties and some applications

In the previous section we saw how to add two vectors and how to multiply a vector by a scalar, but we did not consider how to multiply two vectors. There are two different ways in which we can multiply two vectors, known as the dot product (or scalar product) and the vector product. They are given these names because the result of the first is a scalar and the result of the second is a vector. (We shall not consider vector products in this course.)

In the audio sec
Author(s): The Open University

2.4 Components and the arithmetic of vectors

We introduce now a different method of representing vectors, which will make the manipulation of vectors much easier. Thus we shall avoid having to solve problems involving vectors by drawing the vectors and making measurements, which is very time-consuming and never very accurate.

We can think of a vector as a translation, that is, as representing a movement by a certain amount in a given direction. Then we can use the Cartesian axes in the plane or in Author(s): The Open University

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
Author(s): The Open University

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
Author(s): The Open University

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
Author(s): The Open University

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.

• 
  Author(s): The Open University
  
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4.2.1 ATM physical layer

The ATM physical layer is divided into two sub-layers: the transmission convergence sub-layer and the physical medium sub-layer.

Functions of the transmission convergence sub-layer include generating and receiving cells, and generating and verifying the cyclic redundancy check in the header error control field. For correct interpretation of ATM cells it is important to identify the beginning of a cell. In theory, if ATM cells are transmitted as a continuous stream of bits,
Author(s): The Open University

1.1 Introduction

An interesting analysis of Napoleon's involvement in Spain is provided by Stendhal in A Life of Napoleon, chapters 36 to 43. Stendhal argues that Napoleon's basic error was to see Spain as susceptible to the imposition by the French of the kind of enlightened reforms which had been welcomed elsewhere in Europe. Stendhal particularises, in a way characteristic of Romantic writers, on what he considers a highly distinctive Spanish national character, which in his view explains the hostil
Author(s): The Open University

2.1.3 Concept cards

Another way to tackle unfamiliar words is to start a ‘concept card’ system, using index cards. When you meet a word which seems important, take a new card and write the word at the top, followed by any useful information you have found. File the cards alphabetically and add details as you come across new information. (It is worth getting an index card box anyway, then you can try out various ways of using it to organise your studies.)

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