Calculus: Analytic Geometry and Calculus, with Vectors

2.4 Planes and lines in E3 83

14 Determine the value of the parameter X for which the two planes which
have the equations

2x + 3y + 4z + 5 = 0

2x - 3y - Xz - 5 = 0

are orthogonal. Hint: Modest experiments with two sheets of paper enable us

to capture or recapture the idea that two planes are orthogonal (or normal or

perpendicular to each other) if and only if their normals are orthogonal. .4ns.:

15 If B, C, D are constants for which B and C are not both 0, then the equation

By + Cz + D = 0 is the equation of a plane r. Show that the vector Vl with

scalar components 0, B, C is normal to r. Show that the vector V2 with scalar

components 1,0,0, is normal to the yz plane. Show that Vl and V2 are per-

pendicular and hence that r is perpendicular to the yz plane.

16 Consider again the equation By + Cz + D = 0 or any other equation

involving y and z but not x. Let us agree (this is an important definition) that a

set Sl in Es is a cylinder parallel to a line L if, whenever it contains a point Po, it

also contains all of the points on the line Lo through Po parallel to L. Use this to

show that the graph of the given equation is a cylinder parallel to the x axis.

Solution: Let Po(xo,yo,zo) be any point on the graph of the given equation. Then

the numbers xo, yo, zo satisfy the equation. Since x does not appear in the equa-

tion, it follows that the numbers x, yo, so satisfy the equation for each x. This

means that all of the points (x,yo,zo) on the line Lo through (xo,yo,zo) parallel to

the x axis lie on the graph. Therefore, the graph is a cylinder parallel to the x

axis.

17 Supposing that B, C, and D are constants for which B and C are not both

0 and D 0 0, show that there is no number x for which the numbers x, 0, 0

satisfy the equation By + Cz + D = 0. What is the geometric significance of

this result?

18 Look at the equations

x - xl y -yl z-Zl

X2 - xl Y2 - y1 z2 - Zl

of the line containing two points P,(xl,yl,%,) andP2(xs,y2,z2) Describe completely

the graph of the equation obtained by deleting one of the members of this equality.

Sketch the three graphs obtained in this way.

19 Problem 12 of Problems 2.29 is of interest here. Solve the problem again

and think about it.

'20 Take a good look at the

(1) X2

X

xl(x -xl) +Y2μyl(Y - Yl) -Z±zl(z-zl) =0.

Supposing that Pl(xl,yi,zl) and P2(x2,y2,z2) are two points for which x2 5 xi,

y2 0 yl, z2 zl and that X and μ are numbers not both zero, tell why(1) is the

equation of a plane containing Pl and P2. Then study the text again and attain

a better understanding of matters relating to (1).