Calculus: Analytic Geometry and Calculus, with Vectors

64 Vectors and geometry in three dimensions

11 Make a sketch showing an x, y, z coordinate system,the sphere S having

the equation x2 + y2 + z2 = 9 and theline L having the equations x = 2, y = 2.

Find the length of the part of L that liesinside S. Hint: Do not depend upon

your figure to obtain precisequantitative information. Find and use the coordi-

nates of the points on the spherefor which x = 2 and y = 2. Ans.: 2.

12 By use of the distance formula, show that theequation of the set of points

P(x,y,z) equidistant from two given pointsPi(xi,yi,zi) and P2(x2,y2,z2) can be

put in the form

(1) (x2 - xi) 1 x - Xl2 'x2J + (Y2 - yl)(Y-YI

2

Y2)

` )\

zi +z2)

Remark: Our official introduction to planes in E3 will come inSection 2.4. Mean-

while, we can observe that if Pi and P2 are distinct points, sothat x2 xi or

Y2 F6 yi or z2 F-` zi, thenthe set mentioned above is a plane and the equation

which we have found is its equation. The equation has the form

(2) fl(x - xo) + B(y - yo) + C(z - zo) = 0,

where 11, B, C are constants not all 0 and (xo,yo,zo) is a point in (or on) the plane.

13 Supposing that .1, B, C, xo, yo, zo are constants for which 4, B, C are not

all 0, show that the equation

(1) f1(x - xo) + B(y - yo) + C(z - zo) = 0

is the equation of a plane. Hint: Taking cognizance of Problem 12, solve the

equations

(2) x2-xi=A, y2-yi=B, z2-zi--C

(3) xi + x2

= x0, yi + Y2= Yo,

Si + z2= zo

2 2 2

to obtain two distinct points Pi and P2 such that the graph of (1) is the setof

points equidistant from Pi and P2.

14 The base of a regular tetrahedron has its center at the origin and has

vertices at the points (2a,0,0), (-a,N/1-3 a,0), (-a,- N/J a,0). The other vertex

is on the positive z axis. Find the coordinates of this other vertex. Check the

result by using the distance formula to determine whether the edges have equal

lengths. Finally, sketch the tetrahedron.

15 Determine whether it is possible to multiply all of the coordinates of the

points (2a,0,0), (-a, a,0), (-a,- a,0), (0,0, a) by the same con-

stant X to obtain new points that are vertices of a regular tetrahedron each edge

of which has length a.

16 A set C of points in Es is called a cone with vertex V if whenever it contains

a point P0 different from V it also contains the whole line through V and Po.

Each of these lines is called a generator of the cone, the ancient idea being that

if it moves in an appropriate way, it will "generate" the cone. A cone is called

the circular cone whose vertex is V. whose axis is the line L, and whose central

angle is a if V is on L and the cone consists of the points on those lines through V