Calculus: Analytic Geometry and Calculus, with Vectors

78 Vectors and geometry in three dimensions

direction and hence that there must be a positive constant A such that ok = Au,,
for each k. If possible, draw at least one substantial conclusion from this.
Tell what we would conclude if the correlation coefficient turned out to be -1.
27 Show how something in the preceding problem can be used to prove that

u,o, + uZOS + + 5 -%/ul+ u4 + u;, 9, + v! + + v;

This is the Schwarz inequality. It is both interesting and useful.

2.4 Planes and lines in E3 Planes are important things, and we must
think about them and the natures of their equations. To start the pro-
ceedings, we can think of the top surface of a flat horizontal sheet of
paper as being a part of a plane T. Let P, be a point in w. A vertical
pencil then represents a vector V which is a normal to the plane. With-
out bothering to decide how the fact is related to this or that set of
postulates and definitions in Euclid geometry, we shall use the fact that
a point P different from Pt lies in w if and only if the vector P1P is hori-
zontal, that is, perpendicular to V. Our next step is to apply this idea to
a plane w, shown schematically in Figure 2.43, which is not necessarily
horizontal. Let V be a vector of positive length which is perpendicular
tow and which runs from the origin to a point (A,B,C) not necessarily in w.
Let PI(xl,yl,zl) be a point in w. A point P then lies in w if and only if

(2.401) 0.

This means that either P = P, or PIP is a vector of positive length which
is perpendicular to Y. Thus a point P(x,y,z) lies in w if and only if

(2.402) [Al + Bj + x,)1 + (y - yi)j + (z - z,)k] = 0

or

(2.41) A(x - xI) + B(y - y,) + C(z - z,) = 0

or

(2.42) Ax+By+Cz+D=0,

where D is the constant defined by D = -Ax, - By, - Cz,. It is easy

Figure 2.43

z

\ v-(A,B,G7 that (2.41) is the equation of a plane which

/\ passes through the point (x,,y,,z,) and is

to remember that the equation of a plane

can always be put in the form (2.42), where

A, B, C, D are constants of which A, B, C are

not all zero. It is not so easy to remember

normal to the vector having scalar com-

ponents A, B, C, but this should be done.

To complete this little story, we must prove

that if A, B, C, D are constants for which