Calculus: Analytic Geometry and Calculus, with Vectors

2.2 Coordinate systems and vectors in E, 61

As in Figure 2.242, let OP be the vector running from the origin 0 to the

point P(x,y,z). The rules for multiplying vectors by scalars and for add-

ing vectors imply that

(2.24) OP= xi + yJ + zk.

The three vectors xi, yJ, and zk are the vector components of the vector OP,

and the three scalars x, y, and z are the scalar components.t We can start

getting acquainted with scalar products by observing that the angle

between a vector and itself is 0, so

IOPI2 = 10151 101 cos 0

=

(xi + yJ + zk)

=x2+y2+Z2

and hence that

(2.241) IOPI =.'s/x2+y2+z2.

This important formula holds whether x, y, and z are positive or not.

In case x, y, and z are all positive, we can give another proof of the

formula by applying the Pythagoras theorem twice to the rectangular

Y

Figure 2.242 Figure 2.243

parallelepiped (or brick) of Figure 2.243. Because the angles OQP and
ORQ are right angles, the Pythagoras theorem gives

UP-12 = 101, + 10,12

= IOR12 + IRQI2 + IQP12

=x2+y2+z2

and (2.241) follows. The same methods give the distance formula

(2.25) IPiP1; = /(x2 - xi)2 + (y2 - y,)' + (s. - st)-

t When physicists talk about the components of a vector, they often mean vector com-
ponents. When mathematicians talk about components, they usually mean scalar com-
ponents. Hence the unqualified term "components" is ambiguous. We will have quite a
bonfire if we burn all the books that tell confusing tales about components and projections
and directed distances.