12VECTORS AND THE GEOMETRY OF SPACEOVERVIEW 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.
12.1
Three-DimensionaL Coordinate SystemsTo 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
z
z = constant
I
(O,y,z)
(x, 0, z)
0 ___ 1
P(x,y,z)
- - - __ (O, y,O)
------'y
y = constantx x = constant(x, y, 0)
FIGURE 12.1 The Cartesian coordinate system is right-handed.
660
12.1 Three-Dimensional Coordinate Systems
661
zxz-plane: y = 0
xy-plane: z = 0 - - - -- ____
\
// /
//
___ yz-plane: x
=
0
:\: (0, 0, 0)I
y
Z yLine x = 2, y = 3
(0,3,0)
x
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
0, y 0, z
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. _
zx 2 + y2The circle= 4,
z= 3
(d) x O,y (e) -1 y
1
/
(f) y = - 2, z = 2
EXAMPLE 2
What points P(x, y, z) satisfy the equations and z = 3?
x
FIGURE 12.4 The circle x 2the plane z=
+ y2
=
4 in
3 (Example 2).
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).
12.1 Three-Dimensional Coordinate Systems
663
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 +
G Y)
x 2 + y2 + Z2 + 3x - 4z + I = 0 (X 2 + 3x) + y2 + (Z2 - 4z) =
-I
+ y2 + (Z2 - 4z +
(-;4 y) -I +=
+
3)2 9 21 (x+ 2 +y2+(z-2)2=-1+ 4 +4=4'
(-;4Y
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 + y2 + z2
> 4 (e) x The exterior of the sphere x 2 + y2 + z2 = 4. 2 + y2 +
z2 = 4, z :5 0 (d) x 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 12.1Geometric Interpretations of EquationsIn Exercises
1-16, give a geometric description of the set of points in
Geometric Interpretations of Inequalities and EquationsIn
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 3. yS.
= 0, 2 x + y22
z = 0=
4. x = I, Y = 0z = 0
17. a. x -2 0
0, y
0,
Z
= 0
b. x
0, y
0,
Z
= 0
4,
6. xl
7.
+z 2 =4, y=O 9. x 2 + y2 + z2 = I, x = 0X
+ y2 = 4, z = 8. y2 + z2 = I, x =
18.. 0 '" x '" 1 c. 0 x 1, 019.a.
y= 0
1,
b. 0 '" x '" I, 0 z 1 b.Xl
0 '" Y s 11
x2
+ y2 + z2+ y2 :s 1,
:s 1Z
+ y2 + z2 >+ y2 :s1,Z
10. x 2
11.12. 13. 14. 15.
16.
+ 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
20. a. x 2C. x 2
b. x 2
=3
+ y2:5
1,
norestrictiononz I, z 0b. x=S
21. a. 1 :sx 2 +y2+z2 :S4b. x 2
+ y2 + z2 :sz = 0Z
22. a. x = y,
y,
no restriction on z
23. a. yb. z =
x 2,y3,
0
b. x
y2,
0
S Z
s 2
24. a. z = 1 - y,X
no restriction onx
=2
662Z
Chapter 12: Vectors and the Geometry of Space
Distance and Spheres in SpaceThe 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) isIPI P21 =
V(X2 - XI)2
+
(Y2 - YI)2
+
(Z2 - ZI)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.
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 giveIPIP212 = IPIBI 2 +
IBP21 2
and
(see Figure 12.5). SoIPIP212 = IPIBI 2 + IBP21 2IPIA
12Substitute IP]BI 2 = IP]AI 2 + IABI2 .
+
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)
isIPIP21 = V(-2 - 2f= =Z
+25
(3 - 1)2
+
(0 - 5)2
V16
+4+I"::j
V45
6.708.
Po(Xo, Yo, zo)
P(x,Y,z)
\
al
I
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 f"
I
,,/ 1--
The Standard Equation for the Sphere of Radius a and Center
(xo,Yo, zo)(x - xofY x
+ (y -
YO)2
+
(z - zO)2
= a2
EXAMPLE 4
Find the center and radius of the sphere x2
FIGURE 12.6 The sphere of radius a centered at the point (xo,Yo,
zo).
+ 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 Space44. P I (3, 4, 5),
45. PI(O, 0, 0), b. y-axis at (0, -I, 0) Sphel'l!s Find the centers
and radii of the spheres in Exercises 47-50. P 2(2, 3, 4) P2(2, -2,
-2)
In Exercises 25-34, describe the given set with a single
equatioo or
with a pair of equations.25. The plaoe perpendicular to theL
46. P I (5, 3, -2), P2(0, 0, 0)
x-axis at (3, 0, 0)
c. z-axis at (0, 0, -2)26. The plaoe througb the point (3, -1,2)
perpendicular to theLX-axisxy-planeb. y-axis
c. z-axis
27. The plaoe througb the point (3, -I, I) parallel to theL
b. yz-plane b. yz-plane b. yz-plane
c. xz-plane c. xz-plane
28. The circle of radius 2 centered at (0, 0, 0) and lying in
theL
xy-plane xy-plane
29. The circle of radius 2 centered at (0, 2, 0) and lying in
theL
+2)' + +(z - 2)' = 8 48. I)' +& tY + + + (z 3)' 25 49. (x -
Vz)' + - Vz)2 + + {y (z Vz)' 50. x+ &+tY + (z-tY47. (xy2
(x -
=
=
2
2
=
c. planey = 2Find equations for the spheres whose centers aud
radii are given in Exercises 51-54.
30. The circle of radius I centered at (-3,4, I) and lying in a
planeparallel to theL
xy-plane
b. yz-plane b. y-axis
c. xz-plaoec. z-axis
Center51. (1,2,3) 52. (0, -1,5) 53.
Radius
31. The line through the point (I, 3, -I) parallel to the
LX-axispoint (0, 2, 0)
Vi424 9
32. The set of points in space equidistant from the origin and
the 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 origin34. 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.
(-I,t,-t)x2
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
35. The slab bounded by the planes z = 0 and z = I
(planesincluded) 36. The solid cube in the frrst octant bounded by
1Ire coordioate plaoes and 1Ire planes x = 2, y = 2, and z = 2
+++ 56. ++ + 57. + + +++ 9 58. 3x + + + 9y2z2 -
6y
8z = 0Y
lx
2 2
2y2
2z2
X
z =
3y2
3z2
2y -
2z =
Theory and Examples 59. Find a formula for the distance from the
point P(x, y, z) to the
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 theorigin
a. x-axis
b. y-axis b. yz-plane
c. z-axis
60. Find a formula for the distance from the point P(x, y, z) to
the
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)
a. xy-plane
c. xz-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.
Distance In Exercises 41-46, fmd the distsnce between points PI
and P2.41. Pl(l, I, I), 43. PI(I, 4, 5),
+(y - + +3)' (z
5)' = 4
P 2(3, 3, 0)P2(4, -2,7)
b. thepoint(O, 7, -5).
42. Pl( -I, 1,5), P2(2, 5, 0)
66. Find the point equidistant from the points (0, 0, 0), (0,4,
0), (3, 0, 0), and (2, 2, -3).
12.2
Vectors
665
12.2
__________________________________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.Terminal
Component FormA 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
AB is called a vector.y
FIGURE 12.7
The directed line segment
BA
_______ D
Cp
o(a) two dimensionsx
o
oE
F
(b) three dllnensions
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.
FIGURE 12.8 The velocity vector of a particle moving aloog a
path 0
TI 1 - TIl - TI_I_i
4.
v2
+ _I_j
V3
V6
33. Find a vector ofmagoi1ude 7 in the directioo ofv = 12i - 5k.
34. Find a vector of magoi1ude 3 in the directioo opposite to the
directionofv = (1/2)i - (1/2)j - (1/2)k.Direction and Midplrints In
Exercises 35--38, find
35'
a. the direction of p-;P, aodb. the midpoint ofline segment PI
P,.48. Consider a 25-N weight suspended by two wires as shown in
the accompaoying figure. If the magoi1udes of vectors F I and F,
are both 75 N, then angles a and fJ are equal. Find a.a
35. PI( -I, I, 5) 36. PI(I, 4, 5) 37. P I (3, 4, 5) 38. PI(O, 0,
0)39. If AB 40. IfAB
P,(2, 5, 0) P,(4, -2,7) P,(2, 3, 4) P,(2, -2, -2)
fl
= i + 4j - 2k aodB is the point (5, 1,3), fmdA. = -7i + 3j +
8kaodAisthepoint(-2,-3,6),fmdB.49. Location A bird flies from its
nest 5 km in the directioo 60' north of east, where it stops to
rest 00 a tree. It then flies 10 km in the directioo due southeast
aod lands atop a telephone pole. Place an xy-coordinate system so
that the origin is the bird's nest, the
Theory and Applications
41. Linear combination Let u = 2i + j. v = i + j, and w = i - j.
Find scalars a and b such that u = av + bw.42. Linear combination
Let u = i - 2j, v = 2i + 3j, aod w = i + j. Write U = UI + U2.
where UI is parallel to v and U2 is parallel to w. (See Exercise
41.) 43. Velocity An airplane is flying in the directioo 25' west
of north at 800 kmfh. Find the compooent form of the velocity of
the air-
x-axis points east, and the y-axis points north.a. At what point
is the tree located?b. At what point is the telephone pole?
plane, assuming that the positive x-axis represents due east and
the positive y-axis represents due north.44. (Continuation of
Example 8.) Wbat speed and directioo should the jetliner in Example
8 have in order for the resultant vector to be 500 mph due east?
45. Consider a lOO-N weight suspended by two wires as shown in the
accompanying figure. Find the magoi1udes and compooents of the
force vectors FI aodF,.
50. Use similar triangles to fmd the coordinates of the point Q
that divides the segment from PI(XIoYI, ZI) to P,(x"y" z,) into two
leogths whose ratio is p/q = r. 51. Medians of a triangle Suppose
that A, B, and C are the corner points of the thin 1riangn1ar plate
of constant density shown here. a. Find the vector from Cto the
midPointM ofsideAB.b. Find the vector from C to the point that lies
lwD-thirds of the way from C to M 00 the median CM.
674
Chapter 12: Vectors and the Geometry of Space53. Let ABCD be a
general, not necessarily planar, quadrilateral in space. Show that
the two segments joining the midpoints of opposite sides of ABCD
bisect each other. (Hint: Show that the segments have the same
midpoint)54. Vectors are drawn from the center of a regular n-sided
polygon in the plane to the vertices of the polygon. Show that the
sum of the vectors is zero. (Hint: What happens to the sum if you
rotate the polygon about its center?)
c. Find the coordinates of the point in which the medians of
I!.AJJC intersect According to Exercise 17, Section 6.6, thispoint
is the center of mass.C(I, I, 3)
c.m.B(l, 3, 0)M
55. Suppose that A, B, and C are vertices of a triangle and that
a, b, and c are, the midpoints of the opposite sides. Show thatAa +
Bb + Co o. 56. Unit veeton io the plane Show that a unit vector in
the plane (cosO)i + (sinO)j,obtainedbyrotating can be expressed as
ui through an angle 0 in the couoterclockwise direction. Esplain
why this form gives every unit vector in the plane.
A(4, 2, 0)
52. Find the vector from the origin to the point of intersection
of the medians of the triangle whose vertices are
A(I,-1,2),
B(2,1,3),
and
C(-1,2,-1).
12.3
The Dot ProductIf a force F is applied to a particle moving
along a path, we often need to know the magnitode of the force in
the direction of motion. Ifv is parallel to the tangent line to the
path atthe point where F is applied, then we want the magnitode ofF
in the direction ofv. Figure 12.19 shows that the scalar quantity
we seek is the length IFI cosO, where 0 is the angle between the
two vectors F and v. In this section we show how to calcoIate
easily the angle between two vectors directly from their
components. A key part of the calcoIation is an expression called
the dot product. Dot products are also called inner or scalar
products because the product results in a scalar, not a vector.
After investigating the dot product, we apply it to finding the
projection of one vector onto another (as displayed in Figure
12.19) and to finding the work done by a constant force acting
through a displacement.
v
Length
IF I cos Ii
FIGURE 12.19 The magnitude of the force F in the direction of
vector v is the length IFI cos 0 oftheprojectioo ofF ontov.
Angle Between VectorsWhen two nonzero vectors u and v are placed
so their initial points coincide, they form an angle 0 of measure 0
,,; 0 ,,; 1f (Figure 12.20). If the vectors do not lie along the
same line, the angle 0 is measured in the plane containing both of
them. If they do lie along the same line, the angle between them is
0 if they point in the same direction and 1f if they point in
opposite directions. The angle 0 is the angle between u and v.
Theorem I gives a fonnula to detennine this angle.
v
THEOREM i-Angle Between Two Vectorsvectors u =(Ulo U2, U3)
The angle 0 between two nonzero and v = (Vlo V2, V3) is given
by
oFIGURE 12.20 The angle between u and v.
_
cos
-I (utVt +
lullvl
U2 V2
+ U3V3)
.
Before proving Theorem 1, we focus attention on the expression
UI VI + U2 V2 + U3 V3 in the calcoIation for O. This expression is
the sum of the products of the corresponding components for the
vectors u and v.
12.3
The Dotproduct
675
DEFINmON The dot product u' v ("u dot v") of vectors u = (Ul,
U2, U3) and v = (Vl, V2, V3) is
EXAMPLE 1 (8) (1,-2,-1)'(-6,2,-3)
= =
(1)(-6)
+ (-2)(2) + (-1)(-3) -6 - 4 + 3 = -7=
(b)
(!i +
3j
+
k)
'(4i -
j
+ 2k)
(!)t4) +
(3)(-1)
+
(1)(2) =
I
The dot prodoct of a pair of two-dimensional vectors is dermed
in a similar fashion:
(Ul, U2)' (Vl, V2) = U1 V1
+ U2V2.
We will see throughoot the remainder of the book that the dot
product is a key tool for many important geometric and physical
calculations in space (and the plane), not just for rmding the
angle between two vectors.Proof of Theorem 1 Applying the law of
cosines (Equation (8), Section 1.3) to the triangle in Figure
12.21, we find that
Iwl2 = lul 2 + Ivl2 - 21ullvl cosO 21ullvl cosO = lul 2 + Ivl 2
- Iw1 2 . lul 2 = (VU12 Ivl 2 = (Vv,' Iwl 2 == =
Law of cosines
Becausew = u - v,thecomponentformofwis(U1 - Vl,U2 - V2,U3 -
V3).SO
FIGURE 12.21 The parallelognun law of addition of vectors gives
w = u - v.
+ ul + ul)' = u,' + ul + uJ' + vl + vl)' = v,' + vl + vi (V(U1 -
V1)2 + (U2 - V2)' + (U3 - V3)2)' (U1 - v,)2 + (U2 - vd + (U3 -
V3)22UtVt
u? -
+
Vt
2
+ ui -
2U2V2
+ vi + ul -
2U3V3
+ vl
and Therefore, 21ullvl cosO = lul 2 + Ivl2 - Iwl2 = 2(U1V1
lullvl cos 0 = U1 V1UtVt
+ U2V2 + U3V3)
+ U2V2 + U3V3+U2'V2
+
U3V3
cosO = Since 0:5 ()
lullvl
0
700
Chapter 12: Vector.; and the Geometry of Space
Exercises 12.6Matchtng Equations with Surfaces In Exercises
1-12, match the equation with the surface it defines.Also, identify
each surface by type (paraboloid, ellipsoid, etc.) The swfaces arc
labeled (a)-{I).k. L
,
1. x:1 +y:1 + 4z2 = 10 3. 9y2 +z:1 = 16 5. x _ y 2_ z 2 7.x2 +2z
2 _ S 9. X=z2_ y :1 11. x:1 + 4z:1 = y2L
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 _ y212. 9x:1
,Drawtng
+ 4y:1 + 2z2
= 36
Sketch the surfaces in Exercises 13-44.
b.
,
CYUNDERS13. X2 +y2=4 15. 14.z=y2-1 16. 4x 2 + y2 _ 36 18. 4x:1
20. 9;t2
ELUPSOIDS17. 9;t2 + y2 + z2 = 9 19. 4x 2 + 9y2 + _ 36
+ 4y:1 + z:1 = 16 + 4y2 + 36z2 _ 36
PARABOLOIDS AND CONES
,.
,
d.
,
21. z=x:1+4y:1 23. 25. X:1+ y 2=z:1
:n.z=S-:x2 -y:1 24.y=I-:x 2 -z2 26. 4x:1 + = 9y2 28.y:1+ z 2_ x
2=1 30. (y'/4) - (.'/4) -
HYPERBOLOIDS27.x:1+ y 2- z :1=1 29. z2_ X:1_ y 2= 1
y
z'
1
HYPERBOUC PARABOLOIDS
,
f
,y
31. y2-:x:1 _ z
ASSORTED33. z= 1 +y2_X:1 35. Y _ _ (x:1 + z2) 37. x 2 +y2_z2 _ 4
39. x 2 +z2 _ 1 41. z = _(X2 + y2) 43.4y 2+ z 2_4x:1=4 34. 4x:1 +
4y:1 = z2 36. 1&2 + 4y2 _ 1
y
..L
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
, ,9
4
by the plane z - c as a function of c. (The area of an
ellipse
,
J.
,
with semiaxesa andbis7rab.) b. Use slices perpendicular to the
z-axis to fmd the volume ofthe ellipsoid in part (a).c. Now find
the volume of the ellipsoid
-+-+- - 1. a:1 b2 c2y
x2
y2
z:1
Does your formula give the volume of a sphere of radius a if a -
b - c?
Chapter 12 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?
Questions to Guide Your Review
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. c. Show that the volume in part (a) is also given by
the formulaV=
h "6 (Ao + 4Am + Ah ),
where Am is the area of the region cut by the hyperboloid from
the plane z = h/2.
z
D Plot the surfaces in Exercises 49-52 over the indicated
domains.you can, rotate the surface into different viewing
positions.49. z = y2, 51. z = x 2 52. z = x 2
Viewing Surfaces
If
-2
x
2,x x
-0.5 2,3,yy y
y
2y y
50. z = 1 - y2,y
-2
-2-3 3 3
23
+ y2, -3 + 2y2 overx x x
a. -3 h. -147. Show that the volume ofthe segment cut from the
paraboloidx2 y2 z -2+ - = a b2 c
3, 1, 2,
-3
-2 -2-1
c. -2 d. -2
21
2,
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
hyperboloidx2 y2 z2 -2+ -2 - = 1 a b c2
COMPUTER EXPLORATIONS Use a CAS to plot the surfaces in
Exercises 53-58. Identify the type of quadric surface from your
graph.
53.
9 + 36=
x2
y2
z2=
1 - 25
54.
9 - 9y2
x2
z2
=
1x2
y2
16
55. 5x 2
z2 - 3y2
56.z2
16
=
1-
9 +zz2 = 0
and the planes z = 0 and z = h, h
>
o.
57.
9 -
x2
1
=
16 + 2
y2
58. y -
V4 -
Chapter
Questions to Guide Your Review10. 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).
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.
702
Chapter 12: Vectors and the Geometry of Space
Chapter
Practice ExercisesIn 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.
Vedor Calculations in Two DimensionsIn Exercises 1-4, let u =
(-3,4) and v = (2, -5). Find (a) the componeot form of the vector
aod (b) its magnitude.
1.3u-4v 3. -2u
2.u+v 4. 5v
2S.u=i+j-k,
v=2i+j+k,
w=-i-2j+3k
26.u=i+j,
v=j,
w=i+j+k
In Exercises 5-8, fmd the component form of the vector.
Lines, Planes, and Distances27. 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.
5. The veetor obtained by rotating (0, I) througb an angle
of21f/3
radians6. 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 10. -i - j
11. Velocityveetorv = (-2sinl)i + (2cosI)jwheol = 1f/2.
29. (2,2,0); x
=
-t, Y
=
t,
Z =
-1Z
+t= I
30. (0,4, I); x = 2 + I, y = 2 + 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
Q(I,3, -I).In Esercises 33 and 34, fmd the distance from the
point to the plane.
12. Velocity vector v = (etcost - etsint)i + (etsint + etcost)j
wheol = In 2.
33. (6,0, -6), 34. (3,0, 10),
x - y = 42x
Vedor Calculations in Three DimensionsExpress the veetors in
Exercises 13 and 14 in terms of their leogtha and directions. 13.
2i - 3j
+
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
+ 6k
14. i
+ 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.In Esercises 17 and 18, fmd lvi,
lui, V'u, U'v, v X u, u X v,
P,Q,andR.37. P(i, -1,2), 38. P(i, 0, 0), Q(2, 1,3), R( -1,2, -
1)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. 17. v = i
+ j u=2i+j-2kIn Exercises 19 and 20, fmd proj. u.
Q(O, 1,0), R(O, 0, I)
18. v = i + j + 2k u=-i-k
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.
19. v=2i+j-k u=i+j-5k
20. u=i-2j
v2z
+
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.,
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
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?
x+2y-2z=5
and
5x-2y-z=0
intersect is parallel to the line
x = -3 + 21, y = 31, z = I + 41.
Chapter 1245. The planes 3xL
Practice ExercisesI)J + zk) - 0
703
+ 6z = I and 2x + 2y - z = 3 intcncct in a Iinc.
L (21 - 3J + 3k)'b. x - 3-t,
x + 2)1 + (y - 1) - 3z
Shaw that the planes are orthogonal.
y - -Ilt,
z - 2-3t
b. Find equatiOlli for the line of intersection.
e. (x
+ 2) + l1(y
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? 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).
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. 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.53. Find a vector of magnitude 2 parallel to the line of
intcncction of theplane8x + 2y +z - I = and x -y + 2z+ 7 = O.
d. (2i - 3j + 3k) X x + 2)i + (y - I)j + zk) = 0e. (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
D),A(2,-l,4)
,
C(l.2, 3)
o
54. Find the point in which the line through the origin
perpendicular totheplane2x - y - z = 4mcctsthep1mc3x - 5y +22' =
6.
yB(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
L the coordinates of D,b. the cosine of the interior angle
atB,
x-axis?57. The lineL:
e. the vector projection of BA. onto iiC,
d. the area of the parallelogram,
x=3+2t,
y=2t,
z=t
c. an equatioo for the plane of the parallelogram.f. the areas
of the orthogonal projections of the panillelogram
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.
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)
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 x - 2y + z + 3 =
0 and 2x - y - z + I = O.
""'D(2. 2. 4).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 65-76.65. x 2 + y2
+ z2 = 4
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 your61. Which of the following are
equations for the plane through the points P(I, I, -I), Q(3, 0, 2),
andR(-2, I, O)?
67.4x 2 +4y 2+ Z l=4 69. z = _(x 2 + y2) 71. X2 +y2=z2 73. X2
+y2_z2=475. 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 Exercisesclockwise 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).
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?
z
I I /
_---I.-----_
BI
(l, 3, 2)
zx
---
----.J!___I /
1/
v:
:I //
I
/
y
-------1//
Ship A
ShipB
(4,0,0)
x
(0,5,0)
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.
-\Nor TO SCALESubmarine
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
b.
HI: H2:
x = 6 + 40t, x = 6 + 11 Ot,
Y = -3 + lOt, Y = -3 + 4t,
z = -3 + 2t 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.
(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
121-
T
-
wcosa sin (a+{3)
Chapter 12 Additional and Advanced Exercisesb. For a fIxed {j
determine the value of a which minimizes the
705
magnitude IT ,I. c. For a fIxed a determine the value of (j
which minimizes the magnitude IT217. Determinants and planes
11. Use vectors to show that the distance from P,(XhY') to the
line ax+by=cisd
lax, + by,
-
Va 2 + b 2
cl
a. ShowthatXI X2 -
12. a. Use vectors to ahow that the distance from P,(XhYh z,) to
the plane Ax + By + Cz DisZt Z2 -
XX
X3 -
x
Y' - Y Y2 - Y y, - Y
zZ
d
VA2+B2+C 2
.
Z3 -
z
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 equationxXlX2
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.
13. a. Show that the distance between the parallel planes Ax +
By + Cz D, and Ax + By + Cz D, isd
YYlY2
ZZlZ2
I I
ID, - D21 IAi + Bj + Ckl .6 and
I
07
X3
Y3
Z3
b. Find the distance between the planes 2x + 3y - z 2x + 3y - z
12.
8. Determinant. and line.
Show that the lines
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.d.
Write equations for the planes that lie parallel to and 5 units
away from the plane x - 2y + z 3. 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.
and
intersect or are parallel if and only ifat Cl
b1-d1b2-
a2 a3
C2 C3
d2
=
O.
b3 -d3
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.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.
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.D
2
2Bp
C
1
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.
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: (uX
v)
X
w
(u'w)v - (vw)u. (u'w)v - (uv)w.
10. In the fIgure here, D is the midpoint of side AB oftriang1e
ABC, and E is onethird of the way between C and B. Use vectors to
prove that F is the midpoint of line segment CD.C
u X (v X w)
VerilY each formula for the following vectors by evaluating its
two sides and coroparing the results. u v w
A
D
B
a. 2i b. i-j+k c. 2i+ j d. i+j-2k
Zj Zi+j-Zk 2i-j+k -i - k
Zk -i+Zj-k i + 2k 2i+4j-2k
706
Chapter 12: Vectors and the Geometry of SpaceShow that if u, v,
w, and r are any 21. Use vectors to prove that
18. Cro.s and dot products
vectors, thenb. u X v - (u'v X i)i + (u'v X j)j + (u'v X k)k
c.(uXv).(wxr)_lu,W
(a' + b')(c' + d') '" (ac + bd)'v - ci
ur
V'WI. v"r
for any four nurohers a, b, c, and d. (Hint: Let u - ai + bj and
+ dj.) Show that dot multipli-
22. Dot multiplication is positive def"utite
cation of vectors is positive definite; that is, show that u .
ufor every vector u and that U' u - 0 if and only ifu - O. 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. 25. Showthat
Ivlu + lulvand Ivlu - lulvare orthogonal.
0
19. Cro and dot products
Prove or disprove the formula
u X (u X (u X v))w - -Iul'u'v X w. 20. By forming the cross
product of two appropriate vectors, derive the trigonometric
identity 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.
