Calculus: Analytic Geometry and Calculus, with Vectors

2.1 Vectors in E, 49

first point to the second point as in Figure 2.111. The purpose of the
arrowhead is to show that the vector runs from P1 to P2. The vector
shown in Figure 2.112 is not P1P2, but is P2P1 The length (or magnitude)
JP1P2I of a vector P,P2 is the length of the line segment upon which it lies,
that is, the distance between the points P, and P2. If P2 and P1 coincide,
that is, P2 = P1, the points do not determine a line but they do determine
the vector P1P2, which has length 0 and which is called the zero vector.

1 P,

Figure 2.11 Figure 2.111 Figure 2.112 Figure 2.113

As indicated in Figure 2.113, vectors are often denoted by boldface
letters which keep us informed that the symbols represent vectors rather
than numbers or chemical elements. Thus we can set u = P1P2 and
v = P,P.. Two nonzero vectors u and v are said to be equal, and we
write u = v when, as in Figure 2.113, they (i) lie on parallel lines, (ii) have
equal lengths, and (iii) have the same (not opposite) directions. Two
zero vectors uo and vo do not have directions, but we say that uo = vo
anyway. If u is a nonzero vector and v is a zero vector, then u P& v.
We use the ordinary 0 (zero) to denote the zero vector; it turns out that
we will not need an arrow or distinctive type face to tell us whether 0 is the
number zero or a vector having length zero. The advice given in a the
footnote on page 41 merits repetition here. Whenever we see F(boldface)
or any other letter that is boldface, we recognize that it is a vector and
imagine that there is an arrow above it so that we, in effect, see the sym-
bols F, u, etcetera, that are made by pencils, pens, and chalk. Thus our
imaginations convert what we see into what we write, and the disadvan-
tage of boldface print has disappeared.
It is both interesting and important to know what is meant by the
product kv of a number (real number or scalar) k and a vector v and by the
sum u + v of two vectors. The definitions will imply validity of the
formula 2u = u + u as well as other useful formulas. In case k = 0 or
v = 0 or both, the product kv is the zero vector, that is, kv = 0. In case
k 96 0 and v ;-d 0, the vector kv is a vector such that (i) v and kv lie on the
same or parallel lines, (ii) the length of kv is Iki jvi, and
(iii) v and kv have the same direction if k > 0 and oppo-
site directions if k < 0. Figure 2.12 shows examples.
This definition implies that if v is a nonzero vector,
then the unit vector (vector one unit long) in the direc- V
tion of v is (1/Ivj)v or v/lvl.

Figure 2.12