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12 VECTORS AND THE GEOMETRY OF SPACE OVERVIEW To apply calculus in many real-world situations and in higher mathematics, we need a mathematical description of three-dimensional space. In this chapter we introduce three-dimensional coordinate systems and vectors. Building on what we already know about coordinates in the xy-plane, we establish coordinates in space by adding a third axis that measures distance above and below the xy-plane. Vectors are used to study the analytic geometry of space, where they give simple ways to describe lines, planes, surfaces, and curves in space. We use these geometric ideas later in the book to study motion in space and the calculus of functions of several variables, with their many important applications in science, engineering, economics, and higher mathematics.

Three-DimensionaL Coordinate Systems

12.1

z

z = constant

I

(x, 0, z)

01___

(O,y,z)

P(x,y,z)

- - - __ (O, y,O)

x

x = constant

------'y

y = constant

(x, y, 0)

FIGURE 12.1 The Cartesian coordinate system is right-handed.

660

To locate a point in space, we use three mutually perpendicular coordinate axes, arranged as in Figure 12.1. The axes shown there make a right-handed coordinate frame. When you hold your right hand so that the fingers curl from the positive x-axis toward the positive y-axis, your thumb points along the positive z-axis. So when you look down on the xy-plane from the positive direction of the z-axis, positive angles in the plane are measured counterclockwise from the positive x-axis and around the positive z-axis. (In a left-handed coordinate frame, the z-axis would point downward in Figure 12.1 and angles in the plane would be positive when measured clockwise from the positive x-axis. Right-handed and left-handed coordinate frames are not equivalent.) The Cartesian coordinates (x, y, z) of a point P in space are the values at which the planes through P perpendicular to the axes cut the axes. Cartesian coordinates for space are also called rectangular coordinates because the axes that define them meet at right angles. Points on the x-axis have y- and z-coordinates equal to zero. That is, they have coordinates of the form (x, 0,0). Similarly, points on the y-axis have coordinates of the form (O,y, 0), and points on the z-axis have coordinates of the form (0, 0, z). The planes determined by the coordinates axes are the xy-plane, whose standard equation is z = 0; the yz-plane, whose standard equation is x = 0; and the xz-plane, whose standard equation is y = O. They meet at the origin (0, 0, 0) (Figure 12.2). The origin is also identified by simply 0 or sometimes the letter O. The three coordinate planes x = 0, y = 0, and z = 0 divide space into eight cells called octants. The octant in which the point coordinates are all positive is called the first octant; there is no convention for numbering the other seven octants. The points in a plane perpendicular to the x-axis all have the same x-coordinate, this being the number at which that plane cuts the x-axis. The y- and z-coordinates can be any numbers. Similarly, the points in a plane perpendicular to the y-axis have a common y-coordinate and the points in a plane perpendicular to the z-axis have a common z-coordinate. To write equations for these planes, we name the common coordinate's value. The plane x = 2 is the plane perpendicular to the x-axis at x = 2. The plane y = 3 is the plane perpendicular to the y-axis

12.1 Three-Dimensional Coordinate Systems

661

z xz-plane: y = 0

\

/

xy-plane: z = 0 - - - -- ____



//

/

___ yz-plane: x

=

0

/

(0,3,0)

:\ : (0, 0, 0)

Z y

y

I

x

Line x = 2, y = 3

FIGURE 12.2 The planes x = O, y = O,andz = space into eight octants.

°

divide

FIGURE 12.3 The planes x = 2, y = 3, and z = 5 determine three lines through the point (2,3,5).

at y = 3. The plane z = 5 is the plane perpendicular to the z-axis at z = 5. Figure 12.3 shows the planes x = 2, y = 3, and z = 5, together with their intersection point (2, 3, 5). The planes x = 2 and y = 3 in Figure 12.3 intersect in a line parallel to the z-axis. This line is described by the pair of equations x = 2, y = 3. A point (x, y, z) lies on the line if and only if x = 2 and y = 3. Similarly, the line of intersection of the planes y = 3 and z = 5 is described by the equation pair y = 3, z = 5 . This line runs parallel to the x-axis. The line of intersection of the planes x = 2 and z = 5, parallel to the y-axis, is described by the equation pair x = 2, z = 5 . In the following examples, we match coordinate equations and inequalities with the sets of points they define in space.

EXAMPLE 1 (a) z

We interpret these equations and inequalities geometrically.

°

The half-space consisting of the points on and above the xy-plane.

(b) x = -3

(c) z = 0, x

z The circle

x 2 + y2

= 4,

z= 3

/



0, y



(d) x O,y 0, z (e) -1 y 1 (f) y = - 2, z = 2

EXAMPLE 2

°

°

The plane perpendicular to the x-axis at x = - 3 . This plane lies parallel to the yz-plane and 3 units behind it. The second quadrant of the xy-plane. The first octant. The slab between the planes y = -1 and y = 1 (planes included). The line in which the planes y = - 2 and z = 2 intersect. Alternatively, the line through the point (0, -2, 2) parallel to the x-axis. _

What points P(x, y, z) satisfy the equations and

x

FIGURE 12.4 The circle x 2 the plane z

=

3 (Example 2).

+ y2

=

4 in

z = 3?

Solution The points lie in the horizontal plane z = 3 and, in this plane, make up the circle x 2 + y2 = 4 .We call this set of points "the circle x 2 + y2 = 4 in the plane z = 3" or, _ more simply, "the circlex 2 + y2 = 4, z = 3" (Figure 12.4).

663

12.1 Three-Dimensional Coordinate Systems

quadratic as a squared linear expression. Then, from the equation in standard form, read off the center and radius. For the sphere here, we have

(X2

+ 3x +

GY)

x 2 + y2 + Z2 + 3x - 4z + I = 0 (X 2 + 3x) + y2 + (Z2 - 4z) = -I

(-;4 y) -I +

+ y2 + (Z2 - 4z +

(-;4Y

+

=

21 (x+ 23)2 +y2+(z-2)2=-1+ 49 +4=4'

From this standard form, we read that Xo = -3/2, Yo = 0, Zo = 2, and a = v21/2. The centeris (-3/2,0,2). The radius isv21/2. • EXAMPLE 5 Here are some geometric interpretations of inequalities and equations involving spheres. (a) x 2 + y2 + z2 < 4 The interior of the sphere x 2 + y2 + z2 = 4. (h) x 2 + y2 + z2 :5 4 The solid ball bounded by the sphere x 2 + y2 + z2 = 4. Alternatively, the sphere x 2 + y2 + z2 = 4 together with its interior. 2 (e) x + y2 + z2 > 4 The exterior of the sphere x 2 + y2 + z2 = 4. 2 (d) x + y2 + z2 = 4, z :5 0 The lower hemisphere cut from the sphere x 2 +

y2

+ z2

= 4 by the xy-plane (the plane

z = 0) .

•

Just as polar coordinates give another way to locate points in the xy-plane (Section 11.3), alternative coordinate systems, different from the Cartesian coordinate system developed here, exist for three-dimensional space. We examine two of these coordinate systems in Section 15.7.

Exercises 12.1 Geometric Interpretations of Equations

Geometric Interpretations of Inequalities and Equations

In Exercises 1-16, give a geometric description of the set of points in

In Exercises 17-24, describe the sets of points in space whose coordinates satisfY the given inequalities or combinatioos of equatioos aod inequalities.

space whose coordinates satisfY the given pairs of equations. 1. x = 2, Y = 3 2. x = -1, z = 0

= 0, 2 x + y2

3. y

z = 0

S.

=

2

4,

z = 0

+z 2 =4, y=O 9. x 2 + y2 + z2 = I, x = 0

7.

X

11. 12. 13. 14. 15.

16.

+ y2 = 4, z = 8. y2 + z2 = I, x =

6. xl

+ y2 + z2 = 25, Y = -4 x 2 + y2 + (z + 3)' = 25, z = 0 x 2 + (y - 1)2 + z2 = 4, Y = 0 x 2 + y2 = 4, z = Y x 2 + y2 + z2 = 4, Y = x Y = x 2, z = 0 z = y2, X = 1

10. x 2

17. a. x

4. x = I, Y = 0 -2 0



0, y

0,

Z

18.•. 0 '" x '" 1 c. 0 x 1, 0



y

= 0



b. x

1,



+ y2 + z2

:s 1

20. a. x 2

+ y2 :s 1,

Z

+ y2:5

norestrictiononz

a.

C. x 2

1,

= 0



b. 0 '" x '" I, 0 z 1

x2

19.

0, y

b.

Xl

b. x 2

0,

Z

= 0

0 '" Y s 1

+ y2 + z2 >

1

+ y2 :s

=3

1,

Z

21. a. 1 :sx 2 +y2+z2 :S4 b. x 2

+ y2 + z2 :s

22. a. x = y,

23. a. y x 2,

z = 0 Z

24. a. z = 1 - y, b. z =

y3,

X

0

I, z 0 b. x

=

y,

b. x

S

y2,

no restriction onx

=2

no restriction on z

0

S Z

s 2

662

Chapter 12: Vectors and the Geometry of Space

Distance and Spheres in Space

Z

The formula for the distance between two points in the xy-plane extends to points in space.

The Distance Between P1(XhYh Zl) and P 2(X2,Y2, Z2) is

+

IPI P21 = V(X2 - XI)2

x

FIGURE 12.5 We find the distance between PI and P 2 by applying the Pythagorean theorem to the right triangles PIAB and PIBP2.

(Y2 - YI)2

+

(Z2 - ZI)2

Proof We construct a rectangular box with faces parallel to the coordinate planes and the points PI and P2 at opposite comers of the box (Figure 12.5). If A(X2, YI, zd and B(X2, Y2, Zl) are the vertices of the box indicated in the figure, then the three box edges P I A, AB, and BP2 have lengths

Because triangles PIBP2 and PIAB are both right-angled, two applications of the Pythagorean theorem give IPIP212 = IPIBI 2 + IBP21 2

and

(see Figure 12.5). So IPIP212 = IPIBI 2 + IBP21 2 IPIA 12

+

IABI2

+

Substitute IP]BI 2 = IP]AI 2 + IABI2 .

IBP212

IX2 - xl1 2 + IY2 - YI1 2 + IZ2 - zI1 2 =

(X2 - XI)2

+

(Y2 - Ylf

+

(Z2 - zlf

Therefore

• EXAMPLE 3

The distance between P I (2, 1,5) and P2( -2,3,0) is IPIP21 = V(-2 - 2f

Z

Po(Xo, Yo, zo)

\

P(x,Y,z)

al

I

=

V16

=

V45

+4+ I"::j

+

(3 - 1)2

+

(0 - 5)2

25

•

6.708.

We can use the distance formula to write equations for spheres in space (Figure 12.6). A point P(x, y, z) lies on the sphere of radius a centered at Po(xo,Yo, zo) precisely when IPoPI = a or

I

-- I f"

,,/ 1--

The Standard Equation for the Sphere of Radius a and Center (xo,Yo, zo) (x - xof

+ (y -

YO)2

+

(z - zO)2

= a2

Y x

FIGURE 12.6 The sphere of radius a centered at the point (xo,Yo, zo).

EXAMPLE 4

Find the center and radius of the sphere x2

+ y2 + z2 +

3x - 4z

+ 1 = O.

SoLution We find the center and radius of a sphere the way we find the center and radius of a circle: Complete the squares on the X-, Y-, and z-terms as necessary and write each

664

Chapter 12: Vectors and the Geometry of Space

In Exercises 25-34, describe the given set with a single equatioo or

with a pair of equations. 25. The plaoe perpendicular to the L

x-axis at (3, 0, 0)

b. y-axis

c. z-axis

27. The plaoe througb the point (3, -I, I) parallel to the xy-plane

b. yz-plane

c. xz-plane

28. The circle of radius 2 centered at (0, 0, 0) and lying in the L

xy-plane

b. yz-plane

c. xz-plane

29. The circle of radius 2 centered at (0, 2, 0) and lying in the L

xy-plane

b. yz-plane

parallel to the xy-plane

b. yz-plane

Sphel'l!s Find the centers and radii of the spheres in Exercises 47-50.

+2)' + +(z - 2)' = 8 48. I)' +&+tY +(z +3)' 25 49. (x - Vz)' +{y - Vz)2 +(z +Vz)' 50. x+ &+tY +(z-tY 47. (x

y2

(x -

=

2

b. y-axis

2

=

Find equations for the spheres whose centers aud radii are given in Exercises 51-54.

c. xz-plaoe

Center

Radius

31. The line through the point (I, 3, -I) parallel to the

LX-axis

=

c. planey = 2

30. The circle of radius I centered at (-3,4, I) and lying in a plane L

P2(2, -2, -2)

46. P I (5, 3, -2), P2(0, 0, 0)

26. The plaoe througb the point (3, -1,2) perpendicular to the

L

P 2(2, 3, 4)

45. PI(O, 0, 0), b. y-axis at (0, -I, 0)

c. z-axis at (0, 0, -2) LX-axis

44. P I (3, 4, 5),

c. z-axis

32. The set of points in space equidistant from the origin and the point (0, 2, 0)

53.

33. The circle in whicb the plane through the point (I, 1,3) perpendicular to 1Ire z-axis meets the sphere of radius 5 centered at the origin 34. The set of points in space that lie 2 units from 1Ire point (0, 0, I) and, at 1Ire saroe time, 2 units from the point (0, 0, -I) Inequalities to Describe Sets of Points Write inequalities to describe the sets in Exercises 35-40.

35. The slab bounded by the planes z = 0 and z = I (planes

Vi4

51. (1,2,3) 52. (0, -1,5)

2

(-I,t,-t)

4 9

54. (0, -7,0)

7

Find the centers and radii of the spheres in Exercises 55-58. 55. x 2 y2 z2 4x - 4z = 0

+++ 56. ++ + 57. + + +++ 9 58. 3x + + + 9 x2

lx

y2

2 2

z2 -

8z = 0

6y

2y2

2z2

X

3y2

3z2

2y -

Y

z =

2z =

included) 36. The solid cube in the frrst octant bounded by 1Ire coordioate plaoes and 1Ire planes x = 2, y = 2, and z = 2

37. The half-space consisting of the points 00 and below the xy-plane 38. The upper hemisphere of the sphere of radius I centered at the origin

39. The (a) interior and (b) exterior of the sphere of radius I centered at the point (I, I, I) 40. The closed regioo bounded by the spheres of radius I and radius 2 centered at the origin. (Closed means the spheres are to be included. Had we wanted the spheres left out, we would have asked for the open regioo bounded by the spheres. This is analogous to the way we use closed and open to describe intervals: closed means endpoints included, open means endpoints left out. Closed sets include boundaries; open sets leave them out)

Distance In Exercises 41-46, fmd the distsnce between points PI and P2. 41. Pl(l, I, I),

P 2(3, 3, 0)

42. Pl( -I, 1,5), P2(2, 5, 0) 43. PI(I, 4, 5),

P2(4, -2,7)

Theory and Examples 59. Find a formula for the distance from the point P(x, y, z) to the

a. x-axis

c. z-axis

b. y-axis

60. Find a formula for the distance from the point P(x, y, z) to the

a. xy-plane

c. xz-plane

b. yz-plane

61. Find the perimeter of the triangle with vertices A( -1,2, I), B(I, -I, 3), and C(3, 4, 5). 62. Show that the point p(3, I, 2) is equidistant from the points A(2, -1,3) andB(4, 3, I). 63. Find an equation for the set of all points equidistant from the planesy = 3 andy = -\. 64. Find an equation for the set of all points equidistant from the point (0, 0, 2) and the xy-plane. 65. Find the point on the sphere x 2 nearest •• thexy-plane.

+(y - + + 3)'

(z

5)' = 4

b. thepoint(O, 7, -5).

66. Find the point equidistant from the points (0, 0, 0), (0,4, 0), (3, 0, 0), and (2, 2, -3).

12.2

12.2

Vectors

665

__________________________________ Some of the things we measure are determined simply by their magnitudes. To record mass, length, or time, for example, we need only write down a number and name an appropriate unit of measure. We need more information to describe a force, displacement, or velocity. To describe a force, we need to record the direction in which it acts as well as how large it is. To describe a body's displacement, we have to say in what direction it moved as well as how far. To describe a body's velocity, we have to know where the body is headed as well as how fast it is going. In this section we show how to represent thiogs that have both magnitude and direction in the plane or in space.

Component Form

Terminal

FIGURE 12.7

The directed line segment

AB is called a vector.

A quantity such as force, displacement, or velocity is called a vector and is represented by a directed line segment (Fignre 12.7). The arrow points in the direction of the action and its length gives the magnitude of the action in terms of a suitably chosen uuit. For example, a force vector points in the direction in which the force acts and its length is a measure of the force's strength; a velocity vector points in the direction of motion and its length is the speed of the moving object. Figure 12.8 displays the velocity vector v at a specific location for a particle moving along a path in the plane or in space. (This application of vectors is studied in Chapter 13.) y

y B _______ D

A

C p







o













o













F

E

FIGURE 12.9 The four arrows io the plane (directed lioe segments) shown here have the same length and direction. They therefore represeot the same vector, aodwewriteAB = cD = OP = EF.



x

(a) two dimensions

o











(b) three dllnensions

FIGURE 12.8 The velocity vector of a particle moving aloog a path o. The cross-sections in planes perpendicular to the z-axis above and below the xy-plane are hyperbolas. The cross-sections in planes perpendicular to the other axes are parabolas.

Near the origin, the surface is shaped like a saddle or mountain pass. To a person traveling along the surface in the yz-plane the origin looks like a minimum. To a person traveling the xz-plane the origin looks like a maximum. Such a point is called a saddle point of a surface. We will say more about saddle points in Section 14.7. _ Table 12.1 shows graphs of the six basic types of quadric surfaces. Each surface shown is symmetric with respect to the z-axis, but other coordinate axes can serve as well (with appropriate changes to the equation).

12.6

699

Cylinder.; and Quadric Surfaces

TABLE 12.1 Graphs of Quadric Surfaces

,

, Tho PEIobolu _

ill Ibo ",,"pImo

, -T'---, ,, ,,

" ......

, 'Ibe p,nbolu _







____

;"'1110 1f'1Ilmo

,

,

• ELLIPSOID

ELLIPTICAL PARABOLOID

•

,/

)f--r-"

HYPERBOLOID OF ONE SHEET

ELLIPTICAL CONE

,

. .,.

'Ibe e1Hpee -

+- _1

.'f - cV'i

,

,

. HYPERBOLOID OF 1WO SHEETS

HYPERBOLIC PARABOLOID

% C'

.>0

700

Chapter 12: Vector.; and the Geometry of Space

Exercises 12.6 Matchtng Equations with Surfaces In Exercises 1-12, match the equation with the surface it defines.

k.

,

L

Also, identify each surface by type (paraboloid, ellipsoid, etc.) The swfaces arc labeled (a)-{I).

1. x:1 +y:1 + 4z2 = 10 3. 9y2 +z:1 = 16 5. x _ y 2_ z 2 7.x2 +2z 2 _ S

2.z:1+4y 2_4x:1=4 4.y:1+ z 2=X 2 6.x _ _ 2_ 2 z y 8.z 2 +x:1- y 2 _ 1 10. z = _4.1: 2 _ y2

9. X=z2_ y :1 11. x:1 + 4z:1 = y2

12. 9x:1

+ 4y:1 + 2z2

Drawtng

= 36

,

b.

L

, Sketch the surfaces in Exercises 13-44.

CYUNDERS 13. X2 +y2=4 15.

14.z=y2-1 16. 4x 2 + y2 _ 36

ELUPSOIDS 17. 9;t2 + y2 + z2 = 9 19. 4x 2 + 9y2 + _ 36

18. 4x:1 20. 9;t2

+ 4y:1 + z:1 = 16 + 4y2 + 36z2 _ 36

PARABOLOIDS AND CONES

,

,.

,

d.

:n.z=S-:x2 -y:1

21. z=x:1+4y:1 23. 25. X:1+ y 2=z:1

24.y=I-:x 2 -z2 26. 4x:1 + = 9y2

HYPERBOLOIDS 28.y:1+ z 2_ x 2=1 30. (y'/4) - (.'/4) -

27.x:1+ y 2- z :1=1 29. z2_ X:1_ y 2= 1

y

z'



1

HYPERBOUC PARABOLOIDS

••

,

f•

31. y2-:x:1 _ z

,

ASSORTED

y

39.

y

34. 4x:1 + 4y:1 = z2 36. 1&2 + 4y2 _ 1

33. z= 1 +y2_X:1 35. Y _ _ (x:1 + z2) 37. x 2 +y2_z2 _ 4 x2

+z2



_ 1

+ y2) 43.4y 2+ z 2_4x:1=4 41. z = _(X2

..

,

h•



40. 16y2 + 9z:1 _ 4x 2 42.y:1_ x 2_ z 2=1 44.x 2 +y:1=z

Theory and Examples 45. L Express the area.4. of the cross-section cut from. the ellipsoid

, ,

4 9

L

,

J.

,

by the plane z - c as a function of c. (The area of an ellipse with semiaxesa andbis7rab.) b. Use slices perpendicular to the z-axis to fmd the volume of

the ellipsoid in part (a). c. Now find the volume of the ellipsoid

x2

y2

z:1

-+-+- 1. a:1 b2 c2 y

Does your formula give the volume of a sphere of radius a if a - b - c?

Chapter 12

701

h. Express your answer in part (a) in terms of h and the areas Ao and Ah of the regions cut by the hyperboloid from the planes z = 0 andz = h.

46. The barrel shown here is shaped like an ellipsoid with equal pieces cut from the ends by planes perpendicular to the z-axis. The crosssections perpendicular to the z-axis are circular. The barrel is 2h units high. its midsection radius is R, and its end radii are both r. Find a formula for the barrel's volume. Then check two things. First, suppose the sides of the barrel are straightened to turn the barrel into a cylinder of radius R and height 2h. Does your formula give the cylinder's volume? Second, suppose r = 0 and h = R so the barrel is a sphere. Does your formula give the sphere's volume?

z

Questions to Guide Your Review

c. Show that the volume in part (a) is also given by the formula V=

"6h (Ao + 4Am + Ah ),

where Am is the area of the region cut by the hyperboloid from the plane z = h/2.

Viewing Surfaces

D Plot the surfaces in Exercises 49-52 over the indicated domains.

If

you can, rotate the surface into different viewing positions.

-2 x 2,

49. z = y2,

-2 x 2,

50. z = 1 - y2, y

51. z = x 2 52. z = x 2

47. Show that the volume ofthe segment cut from the paraboloid x2 y2 z -2+ - = a b2 c

by the plane z = h equals half the segment's base times its altitude. 48. a. Find the volume of the solid bounded by the hyperboloid x2 y2 z2 -2+ -2- - = 1 a b c2

and the planes z = 0 and z = h, h

Chapter

>

c. -2 d. -2





x



-3 y 3



3

1, 2,

-2



y



3

-2



y



2

2,

-1



y

x





x

-3

3,



-2 y 2

x 3,







1

COMPUTER EXPLORATIONS Use a CAS to plot the surfaces in Exercises 53-58. Identify the type of quadric surface from your graph. x2

53.

y2

9 + 36

55. 5x 2

o.

+ y2, -3 + 2y2 over

a. -3 h. -1

-0.5 y 2

=

9 -

1 - 25

z2 - 3y2

x2

57.

z2 =

y2

1

=

54.

56. z2

16 + 2

x2

z2

9 - 9 y2

16

58. y -

=

1-

=

1-

x2

y2

16

9 +z

V4 -

z2 = 0

Questions to Guide Your Review

1. When do directed line segments in the plane represent the same vector? 2. How are vectors added and subtracted geometrically? Algebraically? 3. How do you find a vector's magnitude and direction? 4. If a vector is multiplied by a positive scalar, how is the result related to the original vector? What if the scalar is zero? Negative? 5. Define the dot product (scalar product) of two vectors. Which algebraic laws are satisfied by dot products? Give examples. When is the dot product of two vectors equal to zero? 6. What geometric interpretation does the dot product have? Give examples. 7. What is the vector projection of a vector u onto a vector v? Give an example of a useful application of a vector projection. 8. Define the cross product (vector product) of two vectors. Which algebraic laws are satisfied by cross products, and which are not? Give examples. When is the cross product of two vectors equal to zero? 9. What geometric or physical interpretations do cross products have? Give examples.

10. What is the determinant formula for calculating the cross product of two vectors relative to the Cartesian i, j, k-coordinate system? Use it in an example. 11. How do you find equations for lines, line segments, and planes in space? Give examples. Can you express a line in space by a single equation? A plane? 12. How do you find the distance from a point to a line in space? From a point to a plane? Give examples. 13. What are box products? What significance do they have? How are they evaluated? Give an example. 14. How do you find equations for spheres in space? Give examples. 15. How do you find the intersection of two lines in space? A line and a plane? Two planes? Give examples. 16. What is a cylinder? Give examples of equations that define cylinders in Cartesian coordinates. 17. What are quadric surfaces? Give examples of different kinds of ellipsoids, paraboloids, cones, and hyperboloids (equations and sketches).

702

Chapter 12: Vectors and the Geometry of Space

Chapter

Practice Exercises

Vedor Calculations in Two Dimensions In Exercises 1-4, let u = (-3,4) and v = (2, -5). Find (a) the componeot form of the vector aod (b) its magnitude.

In Exercises 25 and 26, fmd (a) the area of the parallelogram determined by vectors u and v and (b) the volume of the parallelepiped determined by the vectors u, v, and w.

1.3u-4v

2.u+v

2S.u=i+j-k,

3. -2u

4. 5v

26.u=i+j,

In Exercises 5-8, fmd the component form of the vector.

5. The veetor obtained by rotating (0, I) througb an angle of21f/3

radians 6. The unit vector that makes an angle of 1f/6 radian with the positivex-axis 7. The vector 2 units loog in the directioo 4i - j 8. The veetor 5 units loog in the directioo opposite to the directioo of(3/5)i + (4/5)j Express the vectors in Exercises 9-12 in terms of their lengtha aod directioos. 9. v2i + v2j

12. Velocity vector v = (etcost - etsint)i + (etsint + etcost)j wheol = In 2.

Vedor Calculations in Three Dimensions Express the veetors in Exercises 13 and 14 in terms of their leogtha and directions.

+ 6k

14. i

v=j,

w=-i-2j+3k

w=i+j+k

Lines, Planes, and Distances 27. Suppose that n is DOnnai to a plane and that v is parallel to the plane. Describe how you would fmd a veetor n that is both perpendicular to v and parallel to the plane. 28. Find a veetor in the plane parallel to the line ax + by = c. In Exercises 29 and 30, fmd the distance from the point to the line.

29. (2,2,0); x

=

-t, Y

=

t,

Z =

+t

-1

30. (0,4, I); x = 2 + I, y = 2 + I,

Z

= I

31. Parametrize the line that passes througb the point (I, 2, 3) parallel to the vector v = - 3i + 7k. 32. Parametrize the line segment joining the points P(I, 2, 0) and

10. -i - j

11. Velocityveetorv = (-2sinl)i + (2cosI)jwheol = 1f/2.

13. 2i - 3j

v=2i+j+k,

+ 2j - k

15. Find a vector 2 units long in the direction ofv = 4i - j + 4k. 16. Find a vector 5 units loog in the direction opposite to the directioo ofv = (3/5) i + (4/5)k.

Q(I,3, -I). In Esercises 33 and 34, fmd the distance from the point to the plane.

33. (6,0, -6),

x - y = 4

34. (3,0, 10),

2x

+

3y

+Z

= 2

35. Find an equation for the plane that passes through the point (3, -2, 1) normal to the veetor n = 2i + j + k. 36. Find an equatioo for the plane that passes through the point (-1,6,0) perpendicular to the line x = -I + I,y = 6 - 21, Z = 31. In Exercises 37 and 38, fmd an equatioo for the plane througb points

P,Q,andR. In Esercises 17 and 18, fmd lvi, lui, V'u, U'v, v X u, u X v,

37. P(i, -1,2),

Iv X u I, the angle between v and u, the scalar compooent of u in the direction afv, and the vector projection ofo onto v.

38. P(i, 0, 0),

17. v = i + j

18. v = i + j + 2k

u=2i+j-2k

u=-i-k

In Exercises 19 and 20, fmd proj. u.

19. v=2i+j-k u=i+j-5k

Q(2, 1,3),

R( -1,2, - 1)

Q(O, 1,0), R(O, 0, I)

39. Find the points in which the line x = I + 21, y = -I - I, z = 3t meets the three coordinate planes. 40. Find the point in which the line througb the origin perpendicular to the plane 2x - Y - Z = 4 meets the plane 3x - 5y + 2z = 6. 41. Find the acute angle between the planes x = 7 and x + y = -3.

v2z

20. u=i-2j

v=i+j+k

+

42. Find the acute angle betweeo the planes x + y = I and

y+z= 1. In Exercises 21 and 22, draw coordinate axes and then sketch u, v, and

u X v as vectors at the origin. 21. u=i.,

v=i+j

22. u=i-j,

v=i+j

23. !flvl = 2, Iwl = 3,andthe angie between v andw is 1f/3,fmd Iv - 2wl· 24. For what value or values of a will the vectors u = 2i + 4j - 5k and v = -4i - 8j + ak be parallel?

43. Find pararne1ric equations for the line in which the planes x + 2y + Z = I and x - y + 2z = -8 intersect. 44. Show that the line in which the planes

x+2y-2z=5

and

5x-2y-z=0

intersect is parallel to the line

x = -3 + 21, y = 31, z = I + 41.

Chapter 12 45. The planes 3x L

+ 6z = I and 2x + 2y - z = 3 intcncct in a Iinc.

Shaw that the planes are orthogonal.

L (21 - 3J + 3k)' b. x - 3-t,

b. Find equatiOlli for the line of intersection.

046. Find an equation for the plane that pasSCll through the point (1,2, 3)paraIleltou - 2i + 3j + kandv - i - j + 2k. 47. Is v = 2i - 4j + k related in any !pCcial way to the plane 2x + Y = 5? Give reasons for your answer. 48. The equation.' P-;P = 0 represents through Po normal to D. What set docs the inequality D' PoP> 0 tcpICscn.t?

«x + 2)1 + (y -

y - -Ilt,

+ 2) + l1(y

e. (x

Practice Exercises

703

I)J + zk) - 0

z - 2-3t

- 1) - 3z

d. (2i - 3j + 3k) X «x + 2)i + (y - I)j + zk) = 0 e. (2i - j + 3k) X (-3i + k)·«x + 2)i + (y - I)j + zk)



6l. The puallelogram !hawn here has vertices at A(2, -1,4), B(I, 0, -I), C(I, 2, 3), andD. Find

,

49. Find the distance from the point P(I, 4, 0) to the plane through A(O. o. 0). B(2. o. -1). "'" C(2. -1.0).

D),

50. Find the distance from the point (2, 2, 3) to the plane 2x + 3y + 5z - O. 51. Find a vector parallel to the plane 2x - y - z - 4 and orthogonaltot+J+k.

A(2,-l,4)

51. Find a unit vector orthogonal to A in the plane of B and C if A - 2i - j + k,B - i+ 2j + k,andC - i + j - 2k.

C(l.2, 3)

53. Find a vector of magnitude 2 parallel to the line of intcncction of theplane8x + 2y +z - I = and x -y + 2z+ 7 = O.

o

54. Find the point in which the line through the origin perpendicular totheplane2x - y - z = 4mcctsthep1mc3x - 5y +22' = 6.

y B(l, 0. -1)

55. Find the point in which the line through p(3, 2, I) normal ttl the plane 2x - y + 2z = -2 meets the plane. 56. What angle doc! the line of intcncction of the planca 2x + Y - z = 0 and x + Y + 2z = 0 make with the positive

x-axis?

L the coordinates of D, b. the cosine of the interior angle atB,

e. the vector projection of BA. onto iiC,

57. The line

d. the area of the parallelogram, L:

x=3+2t,

y=2t,

z=t

in1eDcctll the plane x + 3y - z = -4 in a point P. Find the coonIin.atcs of P and find equations for the line in the plane through P perpendicular ttl L.

58. Shaw that for every real number k the plane

x - 2y+ z + 3 + k(2x - y- z + I) = 0

contains the line of intmscction of the planes

c. an equatioo for the plane of the parallelogram. f. the areas of the orthogonal projections of the panillelogram

on the three coord.i.na.tJ; planes. 63. Distmte betweea lineI Find the distance bctwccn the line L, through the points .4(1,0,-1) and B(-I,I,O) and the line L2 through the points C(3, I, -I) andD(4, 5, -2). The distance is ttl be meuured along the line perpendicular In the two lines. FltBt fmd • vector D perpendicular to both lines. Thm project AC onto D. 64. (Continuatio1l o/Exerci.!e 63.) Find the distance between the line through A(4, 0, 2) and B(2, 4, 1) and the line through C(t, 3, 2)

""'D(2. 2. 4).

x - 2y + z + 3 = 0 and 2x - y - z + I = O. Quadric SumCt!!s 59. Find an equation for the plane through A(-2,O, -3) and B(I, -2, I) that lies parallel to the line through

Identify and sketch the surfaces in Exercises 65-76.

C( -2. -13/'. 26/') ""'D(16/'. -13/'. 0). 60. Is the line x = 1 + 2t, y = -2 + 31, z = -5t related in any way to the plane -4x - 6y + 10z - 91 Give reasoos for your

67.4x 2 +4y 2+ Z l=4 69. z = _(x 2 + y2)

""""'.

61. Which of the following are equations for the plane through the points P(I, I, -I), Q(3, 0, 2), andR(-2, I, O)?

65. x 2 + y2 + z2 = 4

71. X2 +y2=z2 73. X2 +y2_z2=4 75. yl_x2_zl= 1

66. x 2 + (y - 1)2 + z2 = I 68. 3fu:l + 91 2 + 4z2 = 36

70. Y = _(Xl

+ zZ)

71. Xl+zZ=yl

74. 4yl+zZ-ob:l =4 76. z2 _x 2 _ y2 = I

704

Chapter 12: Vectors and the Geometry of Space

Chapter

Additional and Advanced Exercises

1. Submarine hunting Two surface ships on maneuvers are trying to determine a submarine's course and speed to prepare for an aircraft intercept. As shown here, ship A is located at (4, 0, 0), whereas ship B is located at (0, 5, 0). All coordinates are given in thousands of feet. Ship A locates the submarine in the direction of the vector 2i + 3j - (I/3)k, and ship B locates it in the direction of the vector 18i - 6j - k. Four minutes ago, the submarine was located at (2, -1, -1/3). The aircraft is due in 20 min. Assuming that the submarine moves in a straight line at a constant speed, to what position should the surface ships direct the aircraft?

clockwise when we look toward the origin fromA. Find the velocity v of the point of the body that is at the position B(l, 3, 2).

z

I

I

z

---

x

Ship A

ShipB

(4,0,0)

x

(0,5,0)

y

_---I.

B

1/

/



----.J!___ I /

-----_

(l, 3, 2)

I

v:

I

:



I

//

-------1//

/



y

5. Consider the weight suspended by two wires in each diagram. Find the magnitudes and components of vectors F I and F 2, and angles a and {3.

a.

-\ Submarine

Nor TO SCALE

2. A helicopter rescue Two helicopters, HI and H 2, are traveling together. At time t = 0, they separate and follow different straight-line paths given by

HI:

x = 6 + 40t,

Y = -3 + lOt,

z = -3 + 2t

H2:

x = 6 + 11 Ot,

Y = -3 + 4t,

z = -3 + t.

Time t is measured in hours and all coordinates are measured in miles. Due to system malfunctions, H2 stops its flight at (446, 13, 1) and, in a negligible amount of time, lands at (446, 13,0). Two hours later, HI is advised of this fact and heads toward H2 at 150 mph. How long will it take HI to reach H2? 3. Torque The operator's manual for the Toro® 21 in. lawnmower says "tighten the spark plug to 15 ft-lb (20.4 N· m)." If you are installing the plug with a lO.5-in. socket wrench that places the center of your hand 9 in. from the axis of the spark plug, about how hard should you pull? Answer in pounds.

b.

(Hint: This triangle is a right triangle.)

6. Consider a weight of w N suspended by two wires in the diagram, where T I and T 2 are force vectors directed along the wires.

a. Find the vectors T I and T 2 and show that their magnitudes are w cos (3 ITII = sin (a + (3) 4. Rotating body The line through the ongm and the point A( 1, 1, 1) is the axis of rotation of a right body rotating with a constant angular speed of3/2 rad/sec. The rotation appears to be

and T

-

121-

wcosa sin (a+{3)

705

Chapter 12 Additional and Advanced Exercises b. For a fIxed {j determine the value of a which minimizes the

magnitude IT ,I. c. For a fIxed a determine the value of (j which minimizes the magnitude IT21· 7. Determinants and planes

X

X2 -

X

X3 -

x

Y' - Y Y2 - Y y, - Y

Zt -

z

Z2 -

Z

Z3 -

z

d



is an equation for the plane through the three noncollinear points Pt(xt, Yh Zt), P2(X2,)I2, Z2), and P3(X3,Y3, Z3). b. What set of points in space is described by the equation

Y

Z

Xl

Yl

Zl

X2

Y2

Z2

X3

Y3

Z3

x

8. Determinant. and line.

d

lax, + by,

-

cl



Va 2 + b 2

•

12. a. Use vectors to ahow that the distance from P,(XhYh z,) to the plane Ax + By + Cz Dis

a. Showthat XI -

11. Use vectors to show that the distance from P,(XhY') to the line ax+by=cis

I

I

VA2+B2+C 2

13. a. Show that the distance between the parallel planes Ax + By + Cz D, and Ax + By + Cz D, is d



07

.

b. Find an equation for the sphere that is tangent to the planes x + Y + z 3 and x + Y + z 9 if the planes 2x - Y 0 and 3x - z 0 pass through the center of the sphere.



ID, - D21 IAi + Bj + Ckl .

b. Find the distance between the planes 2x + 3y - z 2x + 3y - z 12.

I



6 and

c. Find an equation for the plane parallel to the plane 2x - Y + 2z -4 if the point (3, 2, -I) is equidistant from the two planes.

Show that the lines

d. Write equations for the planes that lie parallel to and 5 units away from the plane x - 2y + z 3.

and

14. Prove that four points A, B, C, and D are coplanar (lie in a common plane) if and only if AD . (AB X BC) O. intersect or are parallel if and only if

15. The projection of a veetor on a plane Let P be a plane in space and let v be a vector. The vector projection ofv onto the plane P, projp v, can be defmed informally as follows. Suppose the suo is shining so that its rays are norma1 to the plane P. Then projp v is the "shadow" ofv onto P. If P is the plane x + 2y + 6z 6 and v i + j + k,fmdprojpv.

b1-d1

at

Cl

a2

C2

b2

a3

C3

b3 -d3

d2

-

O.

=

9. Consider a regular tetrahedron of side length 2.

a. Use vectors to fmd the angle 0 formed by the base of the tetrahedron and any one of its other edges.

16. The accompanying figure shows nonzero vectors v, W, and z, with z orthogonal to the line L, and v aod w ma1cing equal angles {j with L. Assuming Iv I Iwi, fmd w in tenns ofv and z.

D

2

p

2

C

1

B

b. Use vectors to fmd the angle 0 formed by any two adjacent faces of the tetrahedron. This angle is commonly referred to as a dihedral angle.

10. In the fIgure here, D is the midpoint of side AB oftriang1e ABC, and E is one·third of the way between C and B. Use vectors to prove that F is the midpoint of line segment CD.

A

C

D

B

17. Triple vector products The triple vector products (u X v) X w and u X (v X w) are usoally not equal, although the formulas for evaluating them from components are similar: (u

w



(u'w)v - (v·w)u.

u X (v X w)



(u'w)v - (u·v)w.

X

v)

X

VerilY each formula for the following vectors by evaluating its two sides and coroparing the results. u v w

a. 2i b. i-j+k c. 2i+ j d. i+j-2k

Zj Zi+j-Zk

Zk -i+Zj-k

2i-j+k -i - k

i + 2k 2i+4j-2k

706

Chapter 12: Vectors and the Geometry of Space

18. Cro.s and dot products

Show that if u, v, w, and r are any

21. Use vectors to prove that

vectors, then



























(a' + b')(c' + d') '" (ac + bd)'

for any four nurohers a, b, c, and d. (Hint: Let u - ai + bj and + dj.)

b. u X v - (u'v X i)i + (u'v X j)j + (u'v X k)k c.(uXv).(wxr)_lu,W

u·r

19. Cro•• and dot products

V'WI. v"r

v - ci

22. Dot multiplication is positive def"utite



for every vector u and that U' u - 0 if and only ifu - O.

Prove or disprove the formula

u X (u X (u X v))·w - -Iul'u'v X w.

Show that dot multipli-

cation of vectors is positive definite; that is, show that u . u 23. Show that Iu + v I '" Iu I + Iv I for any vectors u and v.

24. Show that w - Iv Iu + Iu Iv bisects the angle between u and v. 20. By forming the cross product of two appropriate vectors, derive the trigonometric identity

25. Showthat Ivlu + lulvand Ivlu - lulvare orthogonal.

sin(A - B) - sinAcosB - cosAsinB.

Chapter

Technology Application Projects

Mathematica/Maple Module: Using to Represent Lilies and Find Distances Parts I and II: Learn the advantages of interpretiog lines as vectors. Part ill: Use vectors to f"md the distance from a point to a line. Putting a in DiMensions onto Il 'IWo-DiIIIensional Caln'llS Use the concept of planes in space to obtain a two-dimensional image.

Getting Started in Plotting in 3D Part I: Use the vector def"mition of lines and planes to generste graphs and equations, and to compare different forms for the equations of a single line. Part II: Plot functions that are def"med implicitly.

0