Unit Vectors

A unit vector is a vector that has a magnitude of exactly 1 and points in a particular direction. It lacks both dimension and unit. Its sole purpose is to point— that is, to specify a direction. The unit vectors in the positive directions of the x, y, and z axes are labeled , , and , where the hat ^ is used instead of an overhead arrow as for other vectors (Fig. 3-14). The arrangement of axes in Fig. 3-14 is said to be a right-handed coordinate system. The system remains right-handed if it is rotated rigidly. We use such coordinate systems exclusively in this book.

Unit vectors are very useful for expressing other vectors; for example, we can express  and  of Figs. 3-8 and 3-9 as

Fig. 3-14 Unit vectors , , and  define the directions of a right-handed coordinate system.

These two equations are illustrated in Fig. 3-15. The quantities a_x and a_y are vectors, called the vector components of . The quantities a_x and a_y are scalars, called the scalar components of  (or, as before, simply its components).

As an example, let us write the displacement  of the spelunking team of Sample Problem 3-3 in terms of unit vectors. First, superimpose the coordinate system of Fig. 3-14 on the one shown in Fig. 3-11a. Then the directions of , , and  are toward the east, up, and toward the south, respectively. Thus, displacement  from start to finish is neatly expressed in unit-vector notation as

Here −(2.6 km) is the vector component d_x along the x axis, and −(2.6 km) is the x component d_x.