Calculus: Analytic Geometry and Calculus, with Vectors

70 Vectors and geometryinthree dimensions

The vector in parentheses is then the unit vector in the direction of P1P2,
and its scalar components are the direction cosines of PIP2.
Before introducing coordinate sys-
P,(x1.r1,Z1) tems, we called attention to the fact
P(x,y,z) that a point P lies on the line P1P2 if

0.0- P2(x2,y2z.) and o if there is a scalar A such

that PIP = XPIP2 and

Figure 2.36 op-0P1).

When the points have coordinates as in Figure 2.36, this equation can be

put in the forms

(2.361) (x - xi)i + (y - Yl)j + (z - z,)k

= X(xs - xi)i + X(ys - Yi)3 + X(zs - z1)k

and

(2.362) x - xI = X(x2 - xi), Y - Y1 = X(y2 - y1),

z - zi = X(z2-zi).

In case x2 76 x1, Y2 0 yi, and z2 0 zi, these equations hold for some X if

and only if

(2.37)

x - xi= Y- yi= z- zi

x2 - XI y2 yi z2 - zi

In case x2 = xi, the condition on x is to be replaced by the condition

x = xi, and similar remarks apply to y2 and z2. With this understanding

the equations (2.37) are equations of the IineP1P2, that is, equations that are

satisfied by x, y, z when and only when P(x,y,z) lies on the line. The

numbers x2 - XI, y2 - yi, Z2 - zi are the numerical components of a

vector lying on the line PIP2, and we know how to find the direction

cosines of this vector.

It can be claimed that the equations

(2.38) x - xi=Y - yi -Z_

a b c

zi

do not look like the equations (2.37) of a line, but we can put these equa-

tions in the form

x - xi y- yi _ z - Zi

(x1+a) - xi (y1+b) - yi (zi+c) - zi

which does have the form (2.37). Thus the equations (2.38) are in fact

equations of the line L which passes through the points Pi(xi,yi,zi) and

P2(xI + a, yi + b, zi + c). The numbers a, b, c are the scalar com-

ponents of the vector P1P2 on L, and they determine the direction cosines

of P1P2 in the usual way. The equations (2.37) and (2.38) are called