Calculus: Analytic Geometry and Calculus, with Vectors

2.4 Planes and lines in E, 81

Therefore,

473)

dA(xi-x)+B(yi-y)+C(z,-z)

(2.

V/A2 + B2 + C2

This looks quite simple but, since P is in r, the equation of r shows that

-Ax - By - Cz=D

and we obtain the more useful formula

(2.48) d

= Ax, + By, + Cz, + DI

/A2+B2+ /+1.2

One who must teach his little sister to start with (2.47) and get (2.48) can

cook up a new five-step rule: (i) rub out the "=0"; (ii) put subscripts on

x, y, z; (iii) divide byV,42 + B2 + C2; (iv) stick on absolute-value signs;

and (v) equate the result to d. It is, as a matter of fact, useful to know

when and how it is possible to prepare instructions so explicit that routine

chores can be performed mechanically and can even be performed by

persons and machines unfamiliar with processes by which formulas are

derived and combined to accomplish their purposes.

Problems 2.49

1 Give geometric interpretations of the numbers xo, yo, zo, A, B, C appearing

in the equation A(x - xo) + B(y - yo) + C(z - zo) = 0 of a plane r and be

prepared to repeat the process at any time. Ans.: See text.

2 Write an intrinsic (not depending upon coordinates) equation of the plane

A which contains a given point Po and is normal to a given vector V. Hint: If

P is in 7, then POP must be perpendicular (or normal or orthogonal) to V. Ans.:

0.

3 How can we derive the coordinate equation of Problem 1 from the intrinsic

equation of Problem 2? Ans.: Set V = Al + Bj + Ck and

PoP= (x - xo)l + (y - yo)j + (z - zo)k

so that

A(x - xo) + B(y - yo) + C(z - zo).

4 Write an intrinsic (not depending upon coordinates) formula for the dis-

tance d from a point P, to a plane u which contains a point P and is normal to

a unit vector n. Hint: Construct an appropriate schematic figure and refer to

Figure 2.471 and formula (2.472) only if assistance is needed.

5 In each case, find the (or an) equation of the plane a which contains the

given point and is perpendicular to a vector having. the given scalar components

(or, in other words, perpendicular to a line having the given direction numbers).

(a) (0,0,0); 1, 1, 1 Ans.: x + y + z = 0

(b) (1,1,1); 1, 1, 1 Ans.: x + y + z = 3

(c) (1,2,3); 4, S, 6 Ans.: 4(x - 1) + S(y - 2) + 6(z - 3) = 0

(d) (x,,y,,z,); A, B, C Ans.: A(x - x,) + B(y - y,) + C(z - z,) = 0