In many applications, we often deal with such things as forces and velocities.

To study them, both their magnitudes and directions must be known. In general, a quantity for which we must specify both magnitude and direction is called a vector.

In basic applications, a vector is usually represented by an arrow showing its magnitude and direction, although this was not common before the 1800s. In 1743, the French mathematician d’Alembert published a paper on dynamics in which he used some diagrams, but most of the text was algebraic. In 1788, the French mathematician Lagrange wrote a classic work on Analytical Mechanics, but the text was all algebraic and included no diagrams.

In the 1800s, calculus was used to greatly advance the use of vectors, and in turn, these advancements became very important in further developments in scientific fields such as electromagnetic theory. Also in the 1800s, mathematicians defined a vector more generally and opened up study in new areas of advanced mathematics. A vector is an excellent example of a math concept that came from a basic physics concept.

In addition to studying vectors in this chapter, we also develop methods of solving oblique triangles (triangles that are not right triangles). Although obviously not known in their modern forms, one of these methods, the law of cosines, was known to the Greek astronomer Ptolemy (90–168), and the other method, the law of sines, was known to Islamic mathematicians of the 1100s. In solving oblique triangles, we often use the trig functions of obtuse angles.

Vectors are of great importance in many fields of science and technology, including physics, engineering, structural design, and navigation. The law of sines and law of cosines are also applied in those fields when working with triangles that don’t contain a right angle.