Calculus: Analytic Geometry and Calculus, with Vectors

76 Vectors and geometry in three dimensions

when k = 1 and when k = 2. Remark: In case x2 = x'1, y2 Y1, 22 = zi,

the transforms of the points on L all coincide with the point (xl,yl, zl). In case

x2 0x1 or y2 FA- y' or z2 54 zi, the transforms of the points on L constitute the

line L' containing the distinct points P',(x',,yi,zi) and P2(x2,y2iz2) and, moreover,

the transforms of points on L between Pl and P2 lie on L' between P'1 and P.

21 Without use of a figure, suppose that 0 is a given number for which

0 < 0 < a/2 and find and simplify the condition that numbers x, y, z (not all

zero) must satisfy if the vector

r =xi+yj+zk

makes one of the angles 0 and 7r - 0 with the positive z axis. Ins.:

±IrI Iki cos 0

and z2 = c2(x2 + y2), where c = cot 0.

22 Two distinct (different) points Po and P1i together with a number 6 for

which 0 < 0 < a/2, determine a right circular cone consisting of Po (the vertex)

and those points P for which the vector PPo makes the angle 6 or 7r - 0 with

the vector POP1. Show that the intrinsic equation of the cone is

(p 1.F P)2 = IPoP1I2IP0FI2 cost B.

Supposing that Po, P1, and P have coordinates (xo,yo,zo), (xl,yl,z1), (x,y,z), and

that

(XI-xo)i+(y'-yo)j+(z,-zo)k = Bi+Bj+Ck,

find the coordinate equation of the cone. Ans.:

[.2(x - xo) + B(y - yo) + C(z - zo)]2

= (f12 + B2 + C2)[(x - xo)2 + (y -yo)2 + (z - zo)2] cost 0.

23 The vertex of a right circular cone is at the point (0,0,h), the axis of the

cone is parallel to the vector i + j, and the lines on the cone make the angle a/4

with the axis of the cone. Find and simplify the equation of the cone. .4ns.:

2xy = (z - A)2. Remark: Putting z = 0 shows that the graph in the xy plane

having the equation xy = h2/2 is a conic section, that is, the intersection of a

right circular cone and a plane.

24 Prove that the graphs of the equations

flx2 + B y2 + Cz2 + Dxy + Exz + Fyz = 0

and

x3+xyz+y3 =0

are cones with vertices at the origin. (See Problem 16 of Problems 2.29 for

information about cones.)

Figure 2.395 (^25) Let Pk(xk,yk,zk), k = 1, 2, 3, 4, be four given points
determining two skew (noncoplanar) lines P1P2 and P3P4 as
P P4 in Figure 2.395. Many persons, including some who are not
easily excited, become quite interested in the problem of
determining a point P'(x',y',z') on P1P2 and a point
P"(x",y",z") on P3P4 such that the line P'P" is perpendicu-
Pa P2 lar to both P1P2 and P$P4. Figure 2.395 may seem to be