Calculus: Analytic Geometry and Calculus, with Vectors

2.1 Vectors in E, 51

is called the angle between u and v or the angle which u makes with v or the

angle which v makes with u. Thus angles are numbers.t If u and v have

the same (or opposite) directions, slight modifications of the above con-

struction give s = 0 (or s = ua) and the same formula (2.151) is used to

define 0. In each case we have 0 < s 5 tra, and hence 0 < 0 5 jr.

When working with angles between vectors, we never have to bother with

"negative angles" and "angles greater than straight angles." For

perpendicular vectors, we have 0 = it/2. We, like electronic computers

and some trigonometric tables, use radian measure and seldom bother

with degrees, minutes, and seconds.

The remainder of the text (not problems) of this section gives basic

information about products of vectors. The importance of the material

will be revealed later in this book and by textbooks in other subjects in

pure and applied mathematics. It is not necessary to presume that the

material is difficult. In fact, students who do not have the good fortune

to study this material calmly in mathematics sometimes find that their

teachers in physics and engineering undertake to teach all of it in a few

seconds.

There are two different kinds of elementary products of vectors u and v

that turn out to be interesting and useful. These are the scalar product

(or dot product) defined by the formula

(2.16) Jul IvI cos 0

and the vector product (or cross product) defined by the formula

(2.17) u X V = Jul Jvi sin on.

These formulas will now be discussed. If u = 0 or v = 0 or both, the

angle 0 appearing in the formulas is not determined by u and v, but the

products u X v are defined

to be 0 anyway. Henceforth, we

consider cases in which Jul > 0 and
Jvi > 0, these being the lengths of u
and v. Then, as in Figures 2.18 and
2.181, the two vectors determine an
angle 0 for which 0 5 0 5 r. In case
0 5 0 5 7/2, the number lvi cos 0

Figure 2.18 Figure 2.181

is the length of the projection of the vector v on the vector u and the
scalar product is therefore the product of the length of u and the length
t Dictionaries convey assorted ideas akin to the ideas that an angle is the "enclosed space"
or "corner" or "opening" near the point where two intersecting lines meet. While we need
not expect to be injured by conflicting meanings of the word angle, we can use the term
"geometric angle B" to signify the "opening" between the two vectors of Figure 2.15. The
number B is then a measure of the size of the geometric angle 0, and we have satisfactory
but somewhat awkward terminology.