Calculus: Analytic Geometry and Calculus, with Vectors

2.3 Scalar products, direction cosines, and lines in E3 75

Thus when we know about vectors, we can put the Newton law in the following
much more useful form. There is a constant G such that a particle P1 ofmass
ml at P1 is attracted toward a particle Pz of mass m2 at P2 by the force F defined
by (3). The problem which we originally proposed involved rectangular coordi-
nates, and it should now be completely obvious that the answer to our problem
is
(x2 - xl)1 + (Y2 - Y1)l + (z2 - zi)k
(4) F=Gmlm2[('K2 - x1)2 + (y2 - yl)2 + (z2 - z1)2]'

Remarks: Persons who work with these things normally recognize the fact that

we do not, in our physical world, have "particles" concentrated at points. It is

sometimes possible, however, to obtain useful results from calculations based on

the assumption that particles are concentrated at points. When this assumption

has been made, we can complicate ideas and simplify language by replacing the

concept of "a particle Pi at the point Pl having mass m1" by the concept of

"a point P1 having mass ml." It can be insisted that this linguistic antic should

be explained; otherwise, a serious student of mathematics is entitled to ask where

the postulates of Euclid provide for the possibility that some points can be

heavier than others. Finally, it can be insisted that the formulas (3) and (4)

should not be remembered; it is better to know (2) and to understand the very

simple process by which we use it to obtain the more useful formulas (3) and (4)

whenever we want them.

20 Let Pi(xl,yl,zi) and P2(x2,y2,z2) be two distinct points in E3 and recall

(or prove again) that to each point P(x,y,z) on the line L containing Pl and P2

there corresponds a number X for which

(1) x = x1 + n(x2 - xl), Y = Y2 + X(Y2 - yl), z = z2 + n(z2 - zl)-

While we may not yet know why such matters are important, we can observe

that the equations

x' = a11x + any + a13z + b1

(2) y' = a21x + a22y + a23z + b2

z' = a31x + a32Y + a33z + b3,

in which the a's and b's are constants, provide mathematical machinery into

which we can substitute the coordinates (x,y,z) of a point in E3 and thereby obtain

the coordinates (x',y',z') of a point (usually another point) in E3. It is very

helpful to think of the equations (2) as being a transformer which transforms a

given point P(x,y,z) into a transform (or transformed point) P'(x',y',z'). Thus

the transformer goes to work on P(x,y,z), which engineers and others call an

input, and produces P'(x',y',z'), which they call an output. This problem con-

cerns transforms of points that lie on the given line L. Find, for each X, the

transform P'(x',y',z') of the point P(x,y,z) on L whose coordinates are given by

(1). .dns.: The result can be put in the form

x' = x1 + X(x2 - x1), y' = A + X(Y2 - A), z' = zi + X(z2 - a1),

where

xk = alixk + a12yk + a13zk + b2

Yk = a21xk + a22yk + a23zk + bx

zk = a31xk + a32yk + a33zk + b3