Formulas are important because they describe *general* relationships, rather than *specific* numerical ones. For example, the tins of paint formula applies to *every* wall. To use such a formula you need to substitute specific values for the general terms, as the following examples show.

## Example 8

Do you want to improve your ability to subtract one number from another, especially if decimals are involved, without having to rely on a calculator? This unit will help you get to grips with subtraction and give you some practice in doing it.

You can start with some practice in subtracting small numbers in your head if you want to. Then we will show you how to subtract bigger numbers on paper. Finally we look at how to subtract decimal numbers.

You don’t need to complete the

How do you deal with divisions where there is a remainder and there are no more digits you can carry to? In most cases you will need to express your answer as a decimal number rather than as a whole number plus a remainder.

Take the example of 518 divided by 8.

In several different parts of the world, footprints from prehistoric human civilisations have been found preserved in either sand or volcanic ash. From these tracks it is possible to measure the foot length and the length of the stride. These measurements can be used to estimate both the height of the person who made the footprint and also whether the person was walking or running by using the following three formulas:

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

the focus

*F*lies on the*x*-axis, and has coordinates (*ae*, 0), where*a*> 0;the directrix

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

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.

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

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):

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 2**v** for **c**, since we can think of a journey repre

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

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* = *m*_{1}*x* + *c*_{1} and *y* = *m*_{2}*x* + *c*_{2}, are parallel if and only if they have the same gradient; that is, if and only if *m*_{1} = *m*_{2}. For example, the lines *y* = −2*x* + 7 and

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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*

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SPC Fall 2012 Graduation - 9:30 a.m. Ceremony
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