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.

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,

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

2008.08.22 - Special Young Benefactors Lecture

Our Young Benefactors membership organization held a special panel discussion on how to start, build, and care for a private art collection on 8/22/2008.

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.)

6 Phylogeny and cladistic analysis

In Section 3.3 the point was made that many physiologists consider that desert birds are successful because of their avian physiology, not because of any specific adaptations. While Williams and Tieleman's research on hoopoe larks demonstrated that desert species are capable of flexibility in metabolic rate and evaporative water loss, it suggested that adaptation is important too. The selective advantages of lowered BMR and TEWL for desert birds include reduced energy demand, and lower produc

The Lost Art of Finding Our Way

http://www.hup.harvard.edu/catalog.php?isbn=9780674072824
Long before GPS, Google Earth, and global transit, humans traveled vast distances using only environmental clues and simple instruments. In his new book, THE LOST ART OF FINDING OUT WAY, John Huth asks what is lost when modern technology substitutes for our innate capacity to find our way. Encyclopedic in breadth, weaving together astronomy, meteorology, oceanography, and ethnography, the book puts us in the shoes, ships, and sleds of ear

The architecture of refugee protection

Annual Harrell-Bond Lecture 2012. Lecture by Professor Mariano-Florentino Cuéllar (Co-Director of the Stanford Center for International Security and Cooperation)recorded on 7 November 2012 at the Oxford Museum of Natural History.

21M.220 Early Music (MIT)

This class covers the history of Western music from antiquity until approximately 1680, about 2000 years worth of music. Rather than cover each topic at the same level of depth, we will focus on four topics in particular and glue them together with a broad overview of other topics. The four topics chosen for this term are (1) chant structure, performance, and development; (2) 14th century music of Italy and France; (3) Elizabethan London; and (4) Venice in the Baroque era.
The class will also in

ISS Update: Nutrition Manager Talks About Children's Book "Space Nutrition"

NASA Public Affairs Officer Brandi Dean interviews Scott Smith, Manager of Nutritional Biochemistry at Johnson Space Center, about the children's book he co-authored called "Space Nutrition." The book talks about the history of spaceflight, the various space programs and of course space food.
Smith talks about how the project developed from a newsletter with illustrations to a children's book. In the book he introduces the "scientific method" and highlights current space experiments and various

Lunchtime at Google with Jacques Pépin

Chef Jacques Pépin joins Anjali Kumar for Lunchtime at Google. This event took place on November 15, 2012, as part of the Lunchtime at Google series.

Programming Oldies But Goodies

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R.G. (Dick) Baldwin
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Over the years, I have published a large number of tutorials in the areas of computer programming and DSP. As I have the time to do so, I am converting the more significant of those tutorials […]