Vitruvius. This Roman architect and engineer, whose full name was Marcus Virtuvius Pollio, flourished in the first century B.C. His Ten Books on Architecture, written as a report to the emperor on the state of the art of building design and construction, is believed to be the oldest book on architecture and engineering that has survived. The classic work is often referred to by its author's name rather than by its title.

Early in his First Book, which in modern terminology would be called the opening chapter, Vitruvius lays out the qualities desirable in an engineer:

See Vitruvius, The Ten Books on Architecture, translated by Morris Hicky Morgan (New York: Dover Publications, 1960); see also “Rereading Vitruvius,” American Scientist, November–December 2010, pp. 457–461.

calculators. The prototype of the now-ubiquitous and inexpensive hand-held battery- or solar-powered electronic calculator was produced in 1966 by Texas Instruments engineers Jack S. Kilby, Jerry D. Merryman, and James H. Van Tasse. Its dimensions were 4–1/4 by 6–1/8 by 1–3/4 inches, and it weighed 45 ounces. The technology, created at TI under the code name Cal-Tech, was licensed by the early 1970s. The Pocketronic calculator went on sale in Japan in 1970 for the equivalent of $395, and became available in the United States in 1971 for $345.

In January 1972, the HP 35, the first scientific pocket calculator, was offered to the public by Hewlett-Packard at a retail price of $395. The introduction of this and soon other “scientific calculators” that could handle trigonometric functions as well as the basic addition, subtraction, multiplication, and division of the Pocketronic caused much debate among engineering professors at the time as to whether such calculators gave students who could afford them an unfair advantage over those who could not. The latter had to continue to use a slide rule, of course. There was no resolution of the academic debate as to whether electronic calculators should be banned from exams before the point became moot because the price of calculators dropped to where they were generally considered as affordable as a good slide rule.

Marchant calculator. Before the advent of the digital computer, this electrically powered calculating machine, whose keyboard was suggestive of a large cash register, but with much smaller and more numerous keys, was among the most sophisticated pieces of equipment available for extensive engineering calculations. A working Marchant, with its register that moved back and forth like a typewriter carriage, had a characteristic mechanical sound that was rotary and repetitive. In an article titled “Socioengineering” (The Bridge, Fall 1994, p. 5), the aerospace engineer Norman Augustine (born in 1935) remembered the 1950s, when Marchants were “the revolutionary new electromechanical desktop computers of the day.” He went on to recall:

For a description of the calculating power of similar machines, such as the “hand-operated, electrically driven Friden mechanical calculators” in the 1940s, see Walter G. Vincenti, “Engineering Theory in the Making: Aerodynamic Calculation ‘Breaks the Sound Barrier’,” Technology and Culture, October 1997, p. 834, where he relates how for some problems in transonic flow, “the numerical work for the four solutions took the better part of a year,” whereas “the same could be done today in seconds on an electronic desk-top computer.”

back of the envelope. This phrase refers to the practice of making a rough sketch of a design or making a very preliminary calculation for the purpose of recording an idea, demonstrating the practicality of a scheme, estimating the magnitude of a phenomenon, or communicating the essence of a concept to a colleague or potential client. A “back-of-the-envelope” sketch or calculation is often the result of an idea or question that arises away from a desk or regular workspace, and so whatever is handy is used as the recording medium.

The phrase evidently dates from times when there were few telephones, let alone laptop computers and e-mail, and when hotels did not conveniently put little pads of notepaper on the table beside the bed. A supply of paper was not taken for granted, as the evidence of so many reused diary pages and other palimpsests attests. Indeed, it has even been said that Abraham Lincoln's ”Gettysburg Address” was written on the back of an envelope as he rode the train from Washington to the Pennsylvania battlefield. Other versions have it that Lincoln wrote the speech in pencil on a brown paper bag, metaphorically still the “back of an envelope.” Recall that the speech was only 272 words long.

The back of an envelope was almost always blank and, except for the slight ridges associated with the construction of the envelope, provided a clean and unimpeded surface on which to draw, write, or calculate.

This book has one purpose: to help you understand vectors and tensors so that you can use them to solve problems. If you're like most students, you first encountered vectors when you took a course dealing with mechanics in high school or college. At that level, you almost certainly learned that vectors are mathematical representations of quantities that have both magnitude and direction, such as velocity and force. You may also have learned how to add vectors graphically and by using their components in the x-, y- and z-directions.

That's a fine place to start, but it turns out that such treatments only scratch the surface of the power of vectors. You can harness that power and make it work for you if you're willing to delve a bit deeper – to see vectors not just as objects with magnitude and direction, but rather as objects that behave in very predictable ways when viewed from different reference frames. That's because vectors are a subset of a larger class of objects called “tensors,” which most students encounter much later in their academic careers, and which have been called “the facts of the Universe.” It is no exaggeration to say that our understanding of the fundamental structure of the universe was changed forever when Albert Einstein succeeded in expressing his theory of gravity in terms of tensors.

If you were tracking the main ideas of Chapter 1, you should realize that vectors are representations of physical quantities – they're mathematical tools that help you visualize and describe a physical situation. In this chapter, you can read about a variety of ways to use those tools to solve problems. You've already seen how to add vectors and how to multiply vectors by a scalar (and why such operations are useful); this chapter contains many other “vector operations” through which you can combine and manipulate vectors. Some of these operations are simple and some are more complex, but each will prove useful in solving problems in physics and engineering. The first section of this chapter explains the simplest form of vector multiplication: the scalar product.

Scalar product

Why is it worth your time to understand the form of vector multiplication called the scalar or “dot” product? For one thing, forming the dot product between two vectors is very useful when you're trying to find the projection of one vector onto another. And why might you want to do that? Well, you may be interested in knowing how much work is done by a force acting on an object. The first instinct of many students is to think of work as “force times distance” (which is a reasonable starting point).