Scalar Multiplication

If you have $100 and by some miracle triple your money, you end up with $300. That process in equation form is $100x3=$300. The point here is that, mathematically, you got the $300 by multiplying the $100 by 3, not by $3. The dollar values ($100 and $300) in that example (from Hoffmann 1975) are counterparts to vectors; the factor 3 is a counterpart to a scalar. In other words, multiplying a vector by a scalar (a constant or pure number) yields another vector.
Figures 5.5a and 5.5b show one- and two-dimensional examples, respectively. In general, if a vector E has coordinates x, y, z, and so on, then the relation for multiplying that vector by a scalar b is

Figure 1: Scalar multiplication, bE.
For example, suppose two-dimensional vector E is the point located at (1,2). Multiplying that vector by the scalar 3 results in a new vector three times as long, located at x=3·1=3 and y=3·2=6 (Fig. 1b).