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Mathematics for Physicists by Graham Shaw, Brian R. Martin

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8 Vectors

In previous chapters we have been concerned exclusively with quantities that are completely specified by their magnitude. These are called scalar quantities, or simply scalars. If they have dimensions, then these also must be specified, in appropriate units. Examples of scalars are temperature, electric charge and mass. In physical science one also meets quantities that are specified by both their magnitude (again in appropriate units) and their direction. Provided they obey the particular law of addition specified below, these are called vector quantities, or just vectors. Examples are force, velocity and magnetic field strength. In this chapter we will be concerned with the algebraic manipulation of vectors, their use in co-ordinate geometry and the most elementary aspects of their calculus. In Chapter 12 we will discuss in more detail the calculus of vectors and vector analysis.

8.1 Scalars and vectors

Because vectors depend on both magnitude and direction, a convenient representation of a vector is by a line with the direction indicated by an arrow anywhere along it, often at its end as shown in Figure 8.1a. The vector represented by the line OA is printed in bold face type a (or if hand-written, as or ). The magnitude of a is the length of the line OA and is a ...

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