Example 2

Here are some vectors of type (b), (c), and (d), illustrated in Figure 10.8.2.

(b)

The vector B = sin εi + cos εj has real length but nonreal direction (where s is a positive infinitesimal). B has length one.

However, B is its own unit vector and is not real, so it has nonreal direction.

Figure 10.8.2 (b) real length only:
B = sin εi + cos εj

(c)

The following vectors have nonreal lengths but real directions.

3εi + 4εj, infinitesimal length 5ε,

(6 + 3ε)i + (8 + 4ε)j, finite length 5(2 + ε),

3Hi + 4Hj, infinite length 5H.

All three of these vectors are parallel and have the same real unit vector

U = 3/5 i + 4/5 j.

Figure 10.8.2 (c) real direction only:
3εi + 4εj
3Hi + 4Hj
(6 + 3ε)i + (8+4ε)j

(d)

The vector D = i + εj has nonreal length and nonreal direction. Its length is , and its unit vector is

Figure 10.8.2 (d) neither
D = i + εj

Figure 10.8.2