Download 12.1 Exercises

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CHAPTER 12
VECTORS AND THE GEOMETRY OF SPACE
Comparing this equation with the standard form, we see that it is the equation of a
sphere with center 共⫺2, 3, ⫺1兲 and radius s8 苷 2 s2 .
EXAMPLE 7 What region in ⺢ 3 is represented by the following inequalities?
1 艋 x 2 ⫹ y 2 ⫹ z2 艋 4
z艋0
SOLUTION The inequalities
z
1 艋 x 2 ⫹ y 2 ⫹ z2 艋 4
can be rewritten as
1 艋 sx 2 ⫹ y 2 ⫹ z 2 艋 2
0
1
2
x
y
FIGURE 13
12.1
so they represent the points 共x, y, z兲 whose distance from the origin is at least 1 and at
most 2. But we are also given that z 艋 0, so the points lie on or below the xy-plane.
Thus the given inequalities represent the region that lies between (or on) the spheres
x 2 ⫹ y 2 ⫹ z 2 苷 1 and x 2 ⫹ y 2 ⫹ z 2 苷 4 and beneath (or on) the xy-plane. It is sketched
in Figure 13.
Exercises
1. Suppose you start at the origin, move along the x-axis a
distance of 4 units in the positive direction, and then move
downward a distance of 3 units. What are the coordinates
of your position?
2. Sketch the points 共0, 5, 2兲, 共4, 0, ⫺1兲, 共2, 4, 6兲, and 共1, ⫺1, 2兲
on a single set of coordinate axes.
3. Which of the points A共⫺4, 0, ⫺1兲, B共3, 1, ⫺5兲, and C共2, 4, 6兲
is closest to the yz-plane? Which point lies in the xz-plane?
4. What are the projections of the point (2, 3, 5) on the xy-, yz-,
and xz-planes? Draw a rectangular box with the origin and
共2, 3, 5兲 as opposite vertices and with its faces parallel to the
coordinate planes. Label all vertices of the box. Find the length
of the diagonal of the box.
5. Describe and sketch the surface in ⺢3 represented by the equa-
tion x ⫹ y 苷 2.
6. (a) What does the equation x 苷 4 represent in ⺢2 ? What does
it represent in ⺢ ? Illustrate with sketches.
(b) What does the equation y 苷 3 represent in ⺢3 ? What does
z 苷 5 represent? What does the pair of equations y 苷 3,
z 苷 5 represent? In other words, describe the set of points
共x, y, z兲 such that y 苷 3 and z 苷 5. Illustrate with a sketch.
3
7–8 Find the lengths of the sides of the triangle PQR. Is it a right
triangle? Is it an isosceles triangle?
7. P共3, ⫺2, ⫺3兲,
8. P共2, ⫺1, 0兲,
Q共7, 0, 1兲,
Q共4, 1, 1兲,
R共1, 2, 1兲
R共4, ⫺5, 4兲
1. Homework Hints available at stewartcalculus.com
9. Determine whether the points lie on straight line.
(a) A共2, 4, 2兲, B共3, 7, ⫺2兲, C共1, 3, 3兲
(b) D共0, ⫺5, 5兲, E共1, ⫺2, 4兲, F共3, 4, 2兲
10. Find the distance from 共3, 7, ⫺5兲 to each of the following.
(a) The xy-plane
(c) The xz-plane
(e) The y-axis
(b) The yz-plane
(d) The x-axis
(f ) The z-axis
11. Find an equation of the sphere with center 共1, ⫺4, 3兲 and
radius 5. What is the intersection of this sphere with the
xz-plane?
12. Find an equation of the sphere with center 共2, ⫺6, 4兲 and
radius 5. Describe its intersection with each of the coordinate
planes.
13. Find an equation of the sphere that passes through the point
共4, 3, ⫺1兲 and has center 共3, 8, 1兲.
14. Find an equation of the sphere that passes through the origin
and whose center is 共1, 2, 3兲.
15–18 Show that the equation represents a sphere, and find its
center and radius.
15. x 2 ⫹ y 2 ⫹ z 2 ⫺ 2x ⫺ 4y ⫹ 8z 苷 15
16. x 2 ⫹ y 2 ⫹ z 2 ⫹ 8x ⫺ 6y ⫹ 2z ⫹ 17 苷 0
17. 2x 2 ⫹ 2y 2 ⫹ 2z 2 苷 8x ⫺ 24 z ⫹ 1
18. 3x 2 ⫹ 3y 2 ⫹ 3z 2 苷 10 ⫹ 6y ⫹ 12z
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SECTION 12.2
19. (a) Prove that the midpoint of the line segment from
P1共x 1, y1, z1 兲 to P2共x 2 , y2 , z2 兲 is
冉
x 1 ⫹ x 2 y1 ⫹ y2 z1 ⫹ z2
,
,
2
2
2
冊
(b) Find the lengths of the medians of the triangle with vertices
A共1, 2, 3兲, B共⫺2, 0, 5兲, and C共4, 1, 5兲.
VECTORS
815
the points on L 2 are directly beneath, or above, the points
on L 1.)
(a) Find the coordinates of the point P on the line L 1.
(b) Locate on the diagram the points A, B, and C, where
the line L 1 intersects the xy-plane, the yz-plane, and the
xz-plane, respectively.
z
20. Find an equation of a sphere if one of its diameters has end-
L¡
points 共2, 1, 4兲 and 共4, 3, 10兲.
21. Find equations of the spheres with center 共2, ⫺3, 6兲 that touch
(a) the xy-plane, (b) the yz-plane, (c) the xz-plane.
P
22. Find an equation of the largest sphere with center (5, 4, 9) that
is contained in the first octant.
1
23–34 Describe in words the region of ⺢ represented by the equa3
0
tions or inequalities.
1
23. x 苷 5
24. y 苷 ⫺2
25. y ⬍ 8
26. x 艌 ⫺3
27. 0 艋 z 艋 6
28. z 2 苷 1
29. x 2 ⫹ y 2 苷 4,
z 苷 ⫺1
30. y 2 ⫹ z 2 苷 16
31. x 2 ⫹ y 2 ⫹ z 2 艋 3
32. x 苷 z
33. x 2 ⫹ z 2 艋 9
34. x 2 ⫹ y 2 ⫹ z 2 ⬎ 2z
35–38 Write inequalities to describe the region.
35. The region between the yz-plane and the vertical plane x 苷 5
37. The region consisting of all points between (but not on) the
spheres of radius r and R centered at the origin, where r ⬍ R
38. The solid upper hemisphere of the sphere of radius 2 centered
at the origin
39. The figure shows a line L 1 in space and a second line L 2 ,
which is the projection of L 1 on the xy-plane. (In other words,
L™
y
x
40. Consider the points P such that the distance from P to
A共⫺1, 5, 3兲 is twice the distance from P to B共6, 2, ⫺2兲. Show
that the set of all such points is a sphere, and find its center and
radius.
41. Find an equation of the set of all points equidistant from the
points A共⫺1, 5, 3兲 and B共6, 2, ⫺2兲. Describe the set.
42. Find the volume of the solid that lies inside both of the spheres
36. The solid cylinder that lies on or below the plane z 苷 8 and on
or above the disk in the xy-plane with center the origin and
radius 2
1
x 2 ⫹ y 2 ⫹ z 2 ⫹ 4x ⫺ 2y ⫹ 4z ⫹ 5 苷 0
and
x 2 ⫹ y 2 ⫹ z2 苷 4
43. Find the distance between the spheres x 2 ⫹ y 2 ⫹ z 2 苷 4 and
x 2 ⫹ y 2 ⫹ z 2 苷 4x ⫹ 4y ⫹ 4z ⫺ 11.
44. Describe and sketch a solid with the following properties.
When illuminated by rays parallel to the z-axis, its shadow is a
circular disk. If the rays are parallel to the y-axis, its shadow is
a square. If the rays are parallel to the x-axis, its shadow is an
isosceles triangle.
Vectors
12.2
D
B
u
v
A
C
FIGURE 1
Equivalent vectors
The term vector is used by scientists to indicate a quantity (such as displacement or velocity or force) that has both magnitude and direction. A vector is often represented by an
arrow or a directed line segment. The length of the arrow represents the magnitude of the
vector and the arrow points in the direction of the vector. We denote a vector by printing a
letter in boldface 共v兲 or by putting an arrow above the letter 共 vl兲.
For instance, suppose a particle moves along a line segment from point A to point B.
The corresponding displacement vector v, shown in Figure 1, has initial point A (the tail)
l
and terminal point B (the tip) and we indicate this by writing v 苷 AB. Notice that the vec-
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CHAPTER 12
VECTORS AND THE GEOMETRY OF SPACE
Equating components, we get
ⱍ ⱍ
ⱍ ⱍ
T
sin
50
ⱍ ⱍ
ⱍ T ⱍ sin 32 苷 980
Solving the first of these equations for ⱍ T ⱍ and substituting into the second, we get
T cos 50
sin 32 苷 980
ⱍ T ⱍ sin 50 ⱍ ⱍ
T1 cos 50 T2 cos 32 苷 0
1
2
2
1
1
cos 32
So the magnitudes of the tensions are
ⱍT ⱍ 苷
1
980
⬇ 839 N
sin 50 tan 32 cos 50
T cos 50
ⱍ T ⱍ 苷 ⱍ cosⱍ 32 ⬇ 636 N
1
and
2
Substituting these values in 5 and 6 , we obtain the tension vectors
T1 ⬇ 539 i 643 j
12.2
Exercises
1. Are the following quantities vectors or scalars? Explain.
(a)
(b)
(c)
(d)
T2 ⬇ 539 i 337 j
The cost of a theater ticket
The current in a river
The initial flight path from Houston to Dallas
The population of the world
5. Copy the vectors in the figure and use them to draw the
following vectors.
(a) u v
(c) v w
(e) v u w
(b) u w
(d) u v
(f ) u w v
2. What is the relationship between the point (4, 7) and the
vector 具4, 7典 ? Illustrate with a sketch.
u
3. Name all the equal vectors in the parallelogram shown.
A
B
E
D
w
v
6. Copy the vectors in the figure and use them to draw the
following vectors.
(a) a b
(c) 12 a
(e) a 2b
(b) a b
(d) 3b
(f ) 2b a
C
b
a
4. Write each combination of vectors as a single vector.
l
l
(a) PQ QR
l l
(c) QS PS
l l
(b) RP PS
l l
l
(d) RS SP PQ
7. In the figure, the tip of c and the tail of d are both the midpoint
of QR. Express c and d in terms of a and b.
Q
P
P
b
a
R
S
1. Homework Hints available at stewartcalculus.com
c
R
d
Q
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SECTION 12.2
ⱍ ⱍ ⱍ ⱍ
8. If the vectors in the figure satisfy u 苷 v 苷 1 and
ⱍ ⱍ
u v w 苷 0, what is w ?
32–33 Find the magnitude of the resultant force and the angle it
makes with the positive x-axis.
32.
u
823
VECTORS
y
33.
20 N
y
200 N
w
v
0
300 N
45°
30°
x
60°
0
x
16 N
9–14 Find a vector a with representation given by the directed line
l
l
segment AB. Draw AB and the equivalent representation starting at
the origin.
9. A共1, 1兲,
B共3, 2兲
10. A共4, 1兲,
11. A共1, 3兲,
B共2, 2兲
12. A共2, 1兲,
13. A共0, 3, 1兲,
B共2, 3, 1兲
14. A共4, 0, 2兲,
B共1, 2兲
B共0, 6兲
B共4, 2, 1兲
15–18 Find the sum of the given vectors and illustrate
geometrically.
15. 具1, 4 典 ,
具6, 2 典
16. 具3, 1 典 ,
17. 具3, 0, 1 典 ,
具0, 8, 0 典
18. 具1, 3, 2 典 ,
ⱍ ⱍ
ⱍ
具1, 5 典
具0, 0, 6 典
ⱍ
19–22 Find a b, 2a 3b, a , and a b .
19. a 苷 具5, 12 典 ,
20. a 苷 4 i j,
b 苷 具3, 6 典
b 苷 i 2j
21. a 苷 i 2 j 3 k,
34. The magnitude of a velocity vector is called speed. Suppose
that a wind is blowing from the direction N45 W at a speed of
50 km兾h. (This means that the direction from which the wind
blows is 45 west of the northerly direction.) A pilot is steering
a plane in the direction N60 E at an airspeed (speed in still air)
of 250 km兾h. The true course, or track, of the plane is the
direction of the resultant of the velocity vectors of the plane
and the wind. The ground speed of the plane is the magnitude
of the resultant. Find the true course and the ground speed of
the plane.
35. A woman walks due west on the deck of a ship at 5 km兾h. The
ship is moving north at a speed of 35 km兾h. Find the speed and
direction of the woman relative to the surface of the water.
36. Ropes 3 m and 5 m in length are fastened to a holiday decora-
tion that is suspended over a town square. The decoration has
a mass of 5 kg. The ropes, fastened at different heights, make
angles of 52 and 40 with the horizontal. Find the tension in
each wire and the magnitude of each tension.
b 苷 2 i j 5 k
22. a 苷 2 i 4 j 4 k,
b 苷 2j k
52°
3 m
40°
5 m
23–25 Find a unit vector that has the same direction as the given
vector.
23. 3 i 7 j
24. 具4, 2, 4典
37. A clothesline is tied between two poles, 8 m apart. The line
25. 8 i j 4 k
26. Find a vector that has the same direction as 具2, 4, 2典 but has
length 6.
27–28 What is the angle between the given vector and the positive
is quite taut and has negligible sag. When a wet shirt with a
mass of 0.8 kg is hung at the middle of the line, the midpoint
is pulled down 8 cm. Find the tension in each half of the
clothesline.
38. The tension T at each end of the chain has magnitude 25 N
(see the figure). What is the weight of the chain?
direction of the x-axis?
28. 8 i 6 j
27. i s3 j
37°
37°
29. If v lies in the first quadrant and makes an angle 兾3 with the
ⱍ ⱍ
positive x-axis and v 苷 4, find v in component form.
30. If a child pulls a sled through the snow on a level path with a
force of 50 N exerted at an angle of 38 above the horizontal,
find the horizontal and vertical components of the force.
31. A quarterback throws a football with angle of elevation 40 and
speed 60 ft兾s. Find the horizontal and vertical components of
the velocity vector.
39. A boatman wants to cross a canal that is 3 km wide and wants
to land at a point 2 km upstream from his starting point. The
current in the canal flows at 3.5 km兾h and the speed of his boat
is 13 km兾h.
(a) In what direction should he steer?
(b) How long will the trip take?
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CHAPTER 12
VECTORS AND THE GEOMETRY OF SPACE
40. Three forces act on an object. Two of the forces are at an angle
of 100 to each other and have magnitudes 25 N and 12 N. The
third is perpendicular to the plane of these two forces and has
magnitude 4 N. Calculate the magnitude of the force that
would exactly counterbalance these three forces.
41. Find the unit vectors that are parallel to the tangent line to the
parabola y 苷 x 2 at the point 共2, 4兲.
42. (a) Find the unit vectors that are parallel to the tangent line to
the curve y 苷 2 sin x at the point 共兾6, 1兲.
(b) Find the unit vectors that are perpendicular to the tangent
line.
(c) Sketch the curve y 苷 2 sin x and the vectors in parts (a)
and (b), all starting at 共兾6, 1兲.
43. If A, B, and C are the vertices of a triangle, find
l
l
l
AB BC CA.
44. Let C be the point on the line segment AB that is twice as far
l
l
l
from B as it is from A. If a 苷 OA, b 苷 OB, and c 苷 OC, show
2
1
that c 苷 3 a 3 b.
45. (a) Draw the vectors a 苷 具3, 2 典 , b 苷 具2, 1 典 , and
c 苷 具7, 1 典.
(b) Show, by means of a sketch, that there are scalars s and t
such that c 苷 sa t b.
(c) Use the sketch to estimate the values of s and t.
(d) Find the exact values of s and t.
48. If r 苷 具x, y典 , r1 苷 具x 1, y1 典 , and r2 苷 具x 2 , y2 典 , describe the
ⱍ
ⱍ ⱍ
ⱍ
set of all points 共x, y兲 such that r r1 r r2 苷 k,
where k r1 r2 .
ⱍ
ⱍ
49. Figure 16 gives a geometric demonstration of Property 2 of
vectors. Use components to give an algebraic proof of this
fact for the case n 苷 2.
50. Prove Property 5 of vectors algebraically for the case n 苷 3.
Then use similar triangles to give a geometric proof.
51. Use vectors to prove that the line joining the midpoints of
two sides of a triangle is parallel to the third side and half
its length.
52. Suppose the three coordinate planes are all mirrored and a
light ray given by the vector a 苷 具a 1, a 2 , a 3 典 first strikes the
xz-plane, as shown in the figure. Use the fact that the angle of
incidence equals the angle of reflection to show that the direction of the reflected ray is given by b 苷 具a 1, a 2 , a 3 典 . Deduce
that, after being reflected by all three mutually perpendicular
mirrors, the resulting ray is parallel to the initial ray. (American
space scientists used this principle, together with laser beams
and an array of corner mirrors on the moon, to calculate very
precisely the distance from the earth to the moon.)
z
46. Suppose that a and b are nonzero vectors that are not parallel
and c is any vector in the plane determined by a and b. Give
a geometric argument to show that c can be written as
c 苷 sa t b for suitable scalars s and t. Then give an argument using components.
b
a
47. If r 苷 具x, y, z典 and r0 苷 具x 0 , y0 , z0 典 , describe the set of all
ⱍ
ⱍ
points 共x, y, z兲 such that r r0 苷 1.
12.3
y
x
The Dot Product
So far we have added two vectors and multiplied a vector by a scalar. The question arises:
Is it possible to multiply two vectors so that their product is a useful quantity? One such
product is the dot product, whose definition follows. Another is the cross product, which is
discussed in the next section.
1 Definition If a 苷 具a 1, a 2 , a 3 典 and b 苷 具b1, b2 , b3 典 , then the dot product of a
and b is the number a ⴢ b given by
a ⴢ b 苷 a 1 b1 a 2 b2 a 3 b3
Thus, to find the dot product of a and b, we multiply corresponding components and
add. The result is not a vector. It is a real number, that is, a scalar. For this reason, the dot
product is sometimes called the scalar product (or inner product). Although Definition 1
is given for three-dimensional vectors, the dot product of two-dimensional vectors is defined
in a similar fashion:
具a 1, a 2 典 ⴢ 具b1, b2 典 苷 a 1 b1 a 2 b2
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CHAPTER 12
VECTORS AND THE GEOMETRY OF SPACE
Exercises
12.3
1. Which of the following expressions are meaningful? Which are
meaningless? Explain.
(a) 共a ⴢ b兲 ⴢ c
(c) a 共b ⴢ c兲
(e) a ⴢ b ⫹ c
ⱍ ⱍ
(b) 共a ⴢ b兲c
(d) a ⴢ 共b ⫹ c兲
(f ) a ⴢ 共b ⫹ c兲
21–22 Find, correct to the nearest degree, the three angles of the
triangle with the given vertices.
21. P共2, 0兲,
Q共0, 3兲, R共3, 4兲
22. A共1, 0, ⫺1兲,
ⱍ ⱍ
2–10 Find a ⴢ b.
B共3, ⫺2, 0兲,
C共1, 3, 3兲
2. a 具⫺2, 3典 ,
b 具0.7, 1.2 典
23–24 Determine whether the given vectors are orthogonal,
parallel, or neither.
3. a 具 ⫺2, 3 典 ,
b 具⫺5, 12典
23. (a) a 具⫺5, 3, 7 典 ,
1
4. a 具6, ⫺2, 3典 ,
5. a 具4, 1,
1
4
典,
b 具6, ⫺3, ⫺8 典
6. a 具s, 2s, 3s典 ,
b 具t, ⫺t, 5t典
7. a i ⫺ 2 j ⫹ 3 k ,
8. a 3 i ⫹ 2 j ⫺ k,
9.
10.
b 具6, ⫺8, 2 典
(b) a 具4, 6 典 , b 具⫺3, 2 典
(c) a ⫺i ⫹ 2 j ⫹ 5 k, b 3 i ⫹ 4 j ⫺ k
(d) a 2 i ⫹ 6 j ⫺ 4 k, b ⫺3 i ⫺ 9 j ⫹ 6 k
b 具2, 5, ⫺1 典
24. (a) u 具⫺3, 9, 6 典 ,
v 具4, ⫺12, ⫺8 典
(b) u i ⫺ j ⫹ 2 k, v 2 i ⫺ j ⫹ k
(c) u 具a, b, c典 , v 具⫺b, a, 0 典
b 5i ⫹ 9k
b 4i ⫹ 5k
ⱍ a ⱍ 6, ⱍ b ⱍ 5 , the angle between a and b is 2␲兾3
ⱍ a ⱍ 3, ⱍ b ⱍ s6 , the angle between a and b is 45⬚
12.
u
具2, 1, ⫺1典 , and 具1, x, 0典 is 45⬚.
u
v
P共1, ⫺3, ⫺2兲, Q共2, 0, ⫺4兲, and R共6, ⫺2, ⫺5兲 is right-angled.
26. Find the values of x such that the angle between the vectors
11–12 If u is a unit vector, find u ⴢ v and u ⴢ w.
11.
25. Use vectors to decide whether the triangle with vertices
v
27. Find a unit vector that is orthogonal to both i ⫹ j and i ⫹ k.
28. Find two unit vectors that make an angle of 60⬚ with
v 具3, 4 典 .
w
29–30 Find the acute angle between the lines.
w
13. (a) Show that i ⴢ j j ⴢ k k ⴢ i 0.
(b) Show that i ⴢ i j ⴢ j k ⴢ k 1.
14. A street vendor sells a hamburgers, b hot dogs, and c soft
drinks on a given day. He charges $2 for a hamburger, $1.50
for a hot dog, and $1 for a soft drink. If A 具a, b, c典 and
P 具2, 1.5, 1 典 , what is the meaning of the dot product A ⴢ P ?
29. 2x ⫺ y 3,
3x ⫹ y 7
30. x ⫹ 2y 7,
5x ⫺ y 2
31–32 Find the acute angles between the curves at their points of
intersection. (The angle between two curves is the angle between
their tangent lines at the point of intersection.)
31. y x 2,
y x3
32. y sin x,
y cos x,
0 艋 x 艋 ␲兾2
15–20 Find the angle between the vectors. (First find an exact
expression and then approximate to the nearest degree.)
33–37 Find the direction cosines and direction angles of the vector.
15. a 具4, 3 典 ,
(Give the direction angles correct to the nearest degree.)
b 具2, ⫺1典
16. a 具⫺2, 5 典 ,
b 具5, 12典
17. a 具3, ⫺1, 5典 ,
18. a 具4, 0, 2典 ,
b 具⫺2, 4, 3 典
b 具2, ⫺1, 0 典
19. a 4i ⫺ 3j ⫹ k,
b 2i ⫺ k
20. a i ⫹ 2 j ⫺ 2 k,
b 4i ⫺ 3k
1. Homework Hints available at stewartcalculus.com
33. 具2, 1, 2典
34. 具6, 3, ⫺2 典
35. i ⫺ 2 j ⫺ 3k
36.
37. 具c, c, c典 ,
1
2
i⫹j⫹k
where c ⬎ 0
38. If a vector has direction angles ␣ ␲兾4 and ␤ ␲兾3, find the
third direction angle ␥.
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SECTION 12.3
39– 44 Find the scalar and vector projections of b onto a.
39. a 具⫺5, 12 典 ,
40. a 具1, 4 典 ,
55. Find the angle between a diagonal of a cube and one of its
56. Find the angle between a diagonal of a cube and a diagonal of
b 具2, 3典
one of its faces.
b 具1, 2, 3 典
42. a 具⫺2, 3, ⫺6 典 ,
b 具5, ⫺1, 4 典
43. a 2 i ⫺ j ⫹ 4 k,
b j ⫹ 12 k
44. a i ⫹ j ⫹ k,
831
edges.
b 具4, 6典
41. a 具3, 6, ⫺2 典 ,
THE DOT PRODUCT
57. A molecule of methane, CH 4 , is structured with the four hydro-
bi⫺j⫹k
45. Show that the vector orth a b b ⫺ proj a b is orthogonal to a.
(It is called an orthogonal projection of b.)
gen atoms at the vertices of a regular tetrahedron and the carbon atom at the centroid. The bond angle is the angle formed
by the H— C—H combination; it is the angle between the
lines that join the carbon atom to two of the hydrogen atoms.
Show that the bond angle is about 109.5⬚. [Hint: Take the
vertices of the tetrahedron to be the points 共1, 0, 0兲, 共0, 1, 0兲,
共0, 0, 1兲, and 共1, 1, 1兲, as shown in the figure. Then the centroid
is ( 12 , 12 , 12 ).]
z
H
46. For the vectors in Exercise 40, find orth a b and illustrate by
drawing the vectors a, b, proj a b, and orth a b.
47. If a 具3, 0, ⫺1 典 , find a vector b such that comp a b 2.
H
C
H
y
48. Suppose that a and b are nonzero vectors.
(a) Under what circumstances is comp a b comp b a ?
(b) Under what circumstances is proj a b proj b a?
49. Find the work done by a force F 8 i ⫺ 6 j ⫹ 9 k that moves
an object from the point 共0, 10, 8兲 to the point 共6, 12, 20兲 along
a straight line. The distance is measured in meters and the force
in newtons.
50. A tow truck drags a stalled car along a road. The chain makes
an angle of 30⬚ with the road and the tension in the chain is
1500 N. How much work is done by the truck in pulling the
car 1 km?
H
x
ⱍ ⱍ
ⱍ ⱍ
58. If c a b ⫹ b a, where a, b, and c are all nonzero vectors,
show that c bisects the angle between a and b.
59. Prove Properties 2, 4, and 5 of the dot product (Theorem 2).
60. Suppose that all sides of a quadrilateral are equal in length and
opposite sides are parallel. Use vector methods to show that the
diagonals are perpendicular.
61. Use Theorem 3 to prove the Cauchy-Schwarz Inequality:
ⱍa ⴢ bⱍ 艋 ⱍaⱍⱍbⱍ
51. A woman exerts a horizontal force of 140 N on a crate as
she pushes it up a ramp that is 4 m long and inclined at an
angle of 20⬚ above the horizontal. Find the work done on
the box.
52. Find the work done by a force of 100 N acting in the direction
N50⬚ W in moving an object 5 m due west.
53. Use a scalar projection to show that the distance from a point
P1共x 1, y1兲 to the line ax ⫹ by ⫹ c 0 is
ⱍ ax
1
⫹ by1 ⫹ c
sa 2 ⫹ b 2
ⱍ
Use this formula to find the distance from the point 共⫺2, 3兲 to
the line 3x ⫺ 4y ⫹ 5 0.
54. If r 具x, y, z典, a 具a 1, a 2 , a 3 典 , and b 具b1, b2 , b3 典 , show
that the vector equation 共r ⫺ a兲 ⴢ 共r ⫺ b兲 0 represents a
sphere, and find its center and radius.
62. The Triangle Inequality for vectors is
ⱍa ⫹ bⱍ 艋 ⱍaⱍ ⫹ ⱍbⱍ
(a) Give a geometric interpretation of the Triangle Inequality.
(b) Use the Cauchy-Schwarz Inequality from Exercise 61 to
prove the Triangle Inequality. [Hint: Use the fact that
a ⫹ b 2 共a ⫹ b兲 ⭈ 共a ⫹ b兲 and use Property 3 of the
dot product.]
ⱍ
ⱍ
63. The Parallelogram Law states that
ⱍa ⫹ bⱍ
2
ⱍ
⫹ a⫺b
ⱍ
2
ⱍ ⱍ
2 a
2
ⱍ ⱍ
⫹2 b
2
(a) Give a geometric interpretation of the Parallelogram Law.
(b) Prove the Parallelogram Law. (See the hint in Exercise 62.)
64. Show that if u ⫹ v and u ⫺ v are orthogonal, then the vectors
u and v must have the same length.
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CHAPTER 12
VECTORS AND THE GEOMETRY OF SPACE
where ␪ is the angle between the position and force vectors. Observe that the only component of F that can cause a rotation is the one perpendicular to r, that is, F sin ␪. The
magnitude of the torque is equal to the area of the parallelogram determined by r and F.
ⱍ ⱍ
EXAMPLE 6 A bolt is tightened by applying a 40-N force to a 0.25-m wrench as shown in
Figure 5. Find the magnitude of the torque about the center of the bolt.
SOLUTION The magnitude of the torque vector is
75°
0.25 m
ⱍ ␶ ⱍ 苷 ⱍ r ⫻ F ⱍ 苷 ⱍ r ⱍⱍ F ⱍ sin 75⬚ 苷 共0.25兲共40兲 sin 75⬚
40 N
苷 10 sin 75⬚ ⬇ 9.66 N⭈m
If the bolt is right-threaded, then the torque vector itself is
␶ 苷 ⱍ ␶ ⱍ n ⬇ 9.66 n
FIGURE 5
12.4
where n is a unit vector directed down into the page.
Exercises
1–7 Find the cross product a ⫻ b and verify that it is orthogonal to
both a and b.
1. a 苷 具6, 0, ⫺2 典 ,
b 苷 具0, 8, 0 典
2. a 苷 具1, 1, ⫺1 典 ,
b 苷 具2, 4, 6 典
3. a 苷 i ⫹ 3 j ⫺ 2 k,
4. a 苷 j ⫹ 7 k,
ⱍ
the page or out of the page.
14.
15.
|v|=5
|v|=16
45°
b 苷 ⫺i ⫹ 5 k
|u|=12
|u|=4
120°
b 苷 2i ⫺ j ⫹ 4k
5. a 苷 i ⫺ j ⫺ k,
b 苷 12 i ⫹ j ⫹ 12 k
6. a 苷 t i ⫹ cos t j ⫹ sin t k,
7. a 苷 具t, 1, 1兾t典,
ⱍ
14–15 Find u ⫻ v and determine whether u ⫻ v is directed into
b 苷 i ⫺ sin t j ⫹ cos t k
16. The figure shows a vector a in the xy-plane and a vector b in
ⱍ
8. If a 苷 i ⫺ 2k and b 苷 j ⫹ k, find a ⫻ b. Sketch a, b, and
9–12 Find the vector, not with determinants, but by using proper-
b
ties of cross products.
11. 共 j ⫺ k兲 ⫻ 共k ⫺ i兲
10. k ⫻ 共i ⫺ 2 j兲
a
12. 共i ⫹ j兲 ⫻ 共i ⫺ j兲
13. State whether each expression is meaningful. If not, explain
why. If so, state whether it is a vector or a scalar.
(a) a ⴢ 共b ⫻ c兲
(b) a ⫻ 共b ⴢ c兲
(c) a ⫻ 共b ⫻ c兲
(d) a ⴢ 共b ⴢ c兲
(e) 共a ⴢ b兲 ⫻ 共c ⴢ d兲
(f ) 共a ⫻ b兲 ⴢ 共c ⫻ d兲
1. Homework Hints available at stewartcalculus.com
ⱍ ⱍ
z
a ⫻ b as vectors starting at the origin.
9. 共i ⫻ j兲 ⫻ k
ⱍ ⱍ
the direction of k. Their lengths are a 苷 3 and b 苷 2.
(a) Find a ⫻ b .
(b) Use the right-hand rule to decide whether the components
of a ⫻ b are positive, negative, or 0.
ⱍ
b 苷 具t 2, t 2, 1 典
x
y
17. If a 苷 具2, ⫺1, 3典 and b 苷 具4, 2, 1典, find a ⫻ b and b ⫻ a.
18. If a 苷 具1, 0, 1典, b 苷 具2, 1, ⫺1 典 , and c 苷 具0, 1, 3典, show that
a ⫻ 共b ⫻ c兲 苷 共a ⫻ b兲 ⫻ c.
19. Find two unit vectors orthogonal to both 具3, 2, 1典 and
具⫺1, 1, 0典.
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SECTION 12.4
20. Find two unit vectors orthogonal to both j ⫺ k and i ⫹ j.
21. Show that 0 ⫻ a 苷 0 苷 a ⫻ 0 for any vector a in V3.
THE CROSS PRODUCT
839
40. Find the magnitude of the torque about P if a 240-N force is
applied as shown.
2m
22. Show that 共a ⫻ b兲 ⴢ b 苷 0 for all vectors a and b in V3 .
P
23. Prove Property 1 of Theorem 11.
2m
24. Prove Property 2 of Theorem 11.
25. Prove Property 3 of Theorem 11.
26. Prove Property 4 of Theorem 11.
30°
240 N
27. Find the area of the parallelogram with vertices A共⫺2, 1兲,
B共0, 4兲, C共4, 2兲, and D共2, ⫺1兲.
28. Find the area of the parallelogram with vertices K共1, 2, 3兲,
L共1, 3, 6兲, M共3, 8, 6兲, and N共3, 7, 3兲.
29–32 (a) Find a nonzero vector orthogonal to the plane through
the points P, Q, and R, and (b) find the area of triangle PQR.
29. P共1, 0, 1兲,
Q共⫺2, 1, 3兲,
R共4, 2, 5兲
30. P共0, 0, ⫺3兲,
Q共4, 2, 0兲,
R共3, 3, 1兲
31. P共0, ⫺2, 0兲,
Q共4, 1, ⫺2兲,
32. P共⫺1, 3, 1兲,
Q共0, 5, 2兲,
41. A wrench 30 cm long lies along the positive y-axis and grips a
bolt at the origin. A force is applied in the direction 具0, 3, ⫺4典
at the end of the wrench. Find the magnitude of the force
needed to supply 100 N⭈m of torque to the bolt.
42. Let v 苷 5 j and let u be a vector with length 3 that starts at
the origin and rotates in the xy -plane. Find the maximum and
minimum values of the length of the vector u ⫻ v. In what
direction does u ⫻ v point?
R共5, 3, 1兲
43. If a ⴢ b 苷 s3 and a ⫻ b 苷 具1, 2, 2典, find the angle between a
R共4, 3, ⫺1兲
and b.
44. (a) Find all vectors v such that
33–34 Find the volume of the parallelepiped determined by the
具1, 2, 1 典 ⫻ v 苷 具3, 1, ⫺5典
vectors a, b, and c.
33. a 苷 具6, 3, ⫺1 典 ,
b 苷 具0, 1, 2典 ,
c 苷 具4, ⫺2, 5典
34. a 苷 i ⫹ j ⫺ k,
b 苷 i ⫺ j ⫹ k,
(b) Explain why there is no vector v such that
具1, 2, 1 典 ⫻ v 苷 具3, 1, 5典
c 苷 ⫺i ⫹ j ⫹ k
45. (a) Let P be a point not on the line L that passes through the
35–36 Find the volume of the parallelepiped with adjacent edges
points Q and R. Show that the distance d from the point P
to the line L is
PQ, PR, and PS.
35. P共⫺2, 1, 0兲,
36. P共3, 0, 1兲,
d苷
Q共2, 3, 2兲, R共1, 4, ⫺1兲, S共3, 6, 1兲
Q共⫺1, 2, 5兲,
R共5, 1, ⫺1兲, S共0, 4, 2兲
37. Use the scalar triple product to verify that the vectors
u 苷 i ⫹ 5 j ⫺ 2 k, v 苷 3 i ⫺ j, and w 苷 5 i ⫹ 9 j ⫺ 4 k
are coplanar.
38. Use the scalar triple product to determine whether the points
A共1, 3, 2兲, B共3, ⫺1, 6兲, C共5, 2, 0兲, and D共3, 6, ⫺4兲 lie in the
same plane.
39. A bicycle pedal is pushed by a foot with a 60-N force as
shown. The shaft of the pedal is 18 cm long. Find the
magnitude of the torque about P.
60 N
l
l
where a 苷 QR and b 苷 QP.
(b) Use the formula in part (a) to find the distance from
the point P共1, 1, 1兲 to the line through Q共0, 6, 8兲 and
R共⫺1, 4, 7兲.
46. (a) Let P be a point not on the plane that passes through the
points Q, R, and S. Show that the distance d from P to the
plane is
a ⴢ 共b ⫻ c兲
d苷
a⫻b
ⱍ
47. Show that a ⫻ b
10°
P
ⱍ
ⱍ
ⱍ
l
l
l
where a 苷 QR, b 苷 QS, and c 苷 QP.
(b) Use the formula in part (a) to find the distance from the
point P共2, 1, 4兲 to the plane through the points Q共1, 0, 0兲,
R共0, 2, 0兲, and S共0, 0, 3兲.
ⱍ
70°
ⱍa ⫻ bⱍ
ⱍaⱍ
ⱍ
2
ⱍ ⱍ ⱍbⱍ
苷 a
2
2
⫺ 共a ⴢ b兲2.
48. If a ⫹ b ⫹ c 苷 0, show that
a⫻b苷b⫻c苷c⫻a
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CHAPTER 12
VECTORS AND THE GEOMETRY OF SPACE
54. If v1, v2, and v3 are noncoplanar vectors, let
49. Prove that 共a ⫺ b兲 ⫻ 共a ⫹ b兲 苷 2共a ⫻ b兲.
50. Prove Property 6 of Theorem 11, that is,
k1 苷
a ⫻ 共b ⫻ c兲 苷 共a ⴢ c兲b ⫺ 共a ⴢ b兲c
v2 ⫻ v3
v1 ⴢ 共v2 ⫻ v3 兲
51. Use Exercise 50 to prove that
k3 苷
a ⫻ 共b ⫻ c兲 ⫹ b ⫻ 共c ⫻ a兲 ⫹ c ⫻ 共a ⫻ b兲 苷 0
52. Prove that
冟
aⴢc bⴢc
共a ⫻ b兲 ⴢ 共c ⫻ d兲 苷
aⴢd bⴢd
53. Suppose that a 苷 0.
冟
(a) If a ⴢ b 苷 a ⴢ c, does it follow that b 苷 c ?
(b) If a ⫻ b 苷 a ⫻ c, does it follow that b 苷 c?
(c) If a ⴢ b 苷 a ⴢ c and a ⫻ b 苷 a ⫻ c, does it follow
that b 苷 c?
DISCOVERY PROJECT
k2 苷
v3 ⫻ v1
v1 ⴢ 共v2 ⫻ v3 兲
v1 ⫻ v2
v1 ⴢ 共v2 ⫻ v3 兲
(These vectors occur in the study of crystallography. Vectors
of the form n1 v1 ⫹ n 2 v2 ⫹ n3 v3 , where each n i is an integer,
form a lattice for a crystal. Vectors written similarly in terms of
k1, k 2 , and k 3 form the reciprocal lattice.)
(a) Show that k i is perpendicular to vj if i 苷 j.
(b) Show that k i ⴢ vi 苷 1 for i 苷 1, 2, 3.
1
(c) Show that k1 ⴢ 共k 2 ⫻ k 3 兲 苷
.
v1 ⴢ 共v2 ⫻ v3 兲
THE GEOMETRY OF A TETRAHEDRON
A tetrahedron is a solid with four vertices, P, Q, R, and S, and four triangular faces, as shown in
the figure.
1. Let v1 , v2 , v3 , and v4 be vectors with lengths equal to the areas of the faces opposite the
P
vertices P, Q, R, and S, respectively, and directions perpendicular to the respective faces and
pointing outward. Show that
v1 ⫹ v2 ⫹ v3 ⫹ v4 苷 0
2. The volume V of a tetrahedron is one-third the distance from a vertex to the opposite face,
S
Q
R
times the area of that face.
(a) Find a formula for the volume of a tetrahedron in terms of the coordinates of its vertices
P, Q, R, and S.
(b) Find the volume of the tetrahedron whose vertices are P共1, 1, 1兲, Q共1, 2, 3兲, R共1, 1, 2兲,
and S共3, ⫺1, 2兲.
3. Suppose the tetrahedron in the figure has a trirectangular vertex S. (This means that the
three angles at S are all right angles.) Let A, B, and C be the areas of the three faces that
meet at S, and let D be the area of the opposite face PQR. Using the result of Problem 1,
or otherwise, show that
D 2 苷 A2 ⫹ B 2 ⫹ C 2
(This is a three-dimensional version of the Pythagorean Theorem.)
12.5
Equations of Lines and Planes
A line in the xy-plane is determined when a point on the line and the direction of the line
(its slope or angle of inclination) are given. The equation of the line can then be written
using the point-slope form.
Likewise, a line L in three-dimensional space is determined when we know a point
P0共x 0 , y0 , z0兲 on L and the direction of L. In three dimensions the direction of a line is conveniently described by a vector, so we let v be a vector parallel to L. Let P共x, y, z兲 be an arbitrary point on L and let r0 and r be the position vectors of P0 and P (that is, they have
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CHAPTER 12
12.5
VECTORS AND THE GEOMETRY OF SPACE
Exercises
1. Determine whether each statement is true or false.
(a)
(b)
(c)
(d)
(e)
(f )
(g)
(h)
(i)
( j)
(k)
Two lines parallel to a third line are parallel.
Two lines perpendicular to a third line are parallel.
Two planes parallel to a third plane are parallel.
Two planes perpendicular to a third plane are parallel.
Two lines parallel to a plane are parallel.
Two lines perpendicular to a plane are parallel.
Two planes parallel to a line are parallel.
Two planes perpendicular to a line are parallel.
Two planes either intersect or are parallel.
Two lines either intersect or are parallel.
A plane and a line either intersect or are parallel.
2–5 Find a vector equation and parametric equations for the line.
2. The line through the point 共6, ⫺5, 2兲 and parallel to the
vector 具 1, 3, ⫺ 23 典
3. The line through the point 共2, 2.4, 3.5兲 and parallel to the
vector 3 i ⫹ 2 j ⫺ k
4. The line through the point 共0, 14, ⫺10兲 and parallel to the line
x 苷 ⫺1 ⫹ 2t, y 苷 6 ⫺ 3t, z 苷 3 ⫹ 9t
5. The line through the point (1, 0, 6) and perpendicular to the
plane x ⫹ 3y ⫹ z 苷 5
6–12 Find parametric equations and symmetric equations for the
line.
6. The line through the origin and the point 共1, 2, 3兲
7. The line through the points (0, 2 , 1) and 共2, 1, ⫺3兲
1
8. The line through the points 共1.0, 2.4, 4.6兲 and 共2.6, 1.2, 0.3兲
9. The line through the points 共⫺8, 1, 4兲 and 共3, ⫺2, 4兲
10. The line through 共2, 1, 0兲 and perpendicular to both i ⫹ j
and j ⫹ k
11. The line through 共1, ⫺1, 1兲 and parallel to the line
x⫹2苷 y苷z⫺3
1
2
12. The line of intersection of the planes x ⫹ 2y ⫹ 3z 苷 1
and x ⫺ y ⫹ z 苷 1
13. Is the line through 共⫺4, ⫺6, 1兲 and 共⫺2, 0, ⫺3兲 parallel to the
line through 共10, 18, 4兲 and 共5, 3, 14兲 ?
14. Is the line through 共⫺2, 4, 0兲 and 共1, 1, 1兲 perpendicular to the
line through 共2, 3, 4兲 and 共3, ⫺1, ⫺8兲 ?
15. (a) Find symmetric equations for the line that passes
through the point 共1, ⫺5, 6兲 and is parallel to the vector
具⫺1, 2, ⫺3 典 .
(b) Find the points in which the required line in part (a) intersects the coordinate planes.
1. Homework Hints available at stewartcalculus.com
16. (a) Find parametric equations for the line through 共2, 4, 6兲 that
is perpendicular to the plane x ⫺ y ⫹ 3z 苷 7.
(b) In what points does this line intersect the coordinate
planes?
17. Find a vector equation for the line segment from 共2, ⫺1, 4兲
to 共4, 6, 1兲.
18. Find parametric equations for the line segment from 共10, 3, 1兲
to 共5, 6, ⫺3兲.
19–22 Determine whether the lines L 1 and L 2 are parallel, skew, or
intersecting. If they intersect, find the point of intersection.
19. L 1: x 苷 3 ⫹ 2t,
y 苷 4 ⫺ t, z 苷 1 ⫹ 3t
L 2: x 苷 1 ⫹ 4s, y 苷 3 ⫺ 2s, z 苷 4 ⫹ 5s
20. L 1: x 苷 5 ⫺ 12t,
y 苷 3 ⫹ 9t, z 苷 1 ⫺ 3t
L 2: x 苷 3 ⫹ 8s, y 苷 ⫺6s, z 苷 7 ⫹ 2s
21. L 1:
x⫺2
y⫺3
z⫺1
苷
苷
1
⫺2
⫺3
L 2:
x⫺3
y⫹4
z⫺2
苷
苷
1
3
⫺7
22. L 1:
x
y⫺1
z⫺2
苷
苷
1
⫺1
3
L 2:
x⫺2
y⫺3
z
苷
苷
2
⫺2
7
23– 40 Find an equation of the plane.
23. The plane through the point 共6, 3, 2兲 and perpendicular to the
vector 具⫺2, 1, 5 典
24. The plane through the point 共4, 0, ⫺3兲 and with normal
vector j ⫹ 2 k
25. The plane through the point (⫺1, 2 , 3) and with normal
1
vector i ⫹ 4 j ⫹ k
26. The plane through the point 共2, 0, 1兲 and perpendicular to the
line x 苷 3t, y 苷 2 ⫺ t, z 苷 3 ⫹ 4t
27. The plane through the point 共1, ⫺1, ⫺1兲 and parallel to the
plane 5x ⫺ y ⫺ z 苷 6
28. The plane through the point 共2, 4, 6兲 and parallel to the plane
z苷x⫹y
29. The plane through the point (1, 2 , 3 ) and parallel to the plane
1 1
x⫹y⫹z苷0
30. The plane that contains the line x 苷 1 ⫹ t, y 苷 2 ⫺ t,
z 苷 4 ⫺ 3t and is parallel to the plane 5x ⫹ 2y ⫹ z 苷 1
31. The plane through the points 共0, 1, 1兲, 共1, 0, 1兲, and 共1, 1, 0兲
32. The plane through the origin and the points 共2, ⫺4, 6兲
and 共5, 1, 3兲
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SECTION 12.5
EQUATIONS OF LINES AND PLANES
849
33. The plane through the points 共3, ⫺1, 2兲, 共8, 2, 4兲, and
57–58 (a) Find parametric equations for the line of intersection of
the planes and (b) find the angle between the planes.
34. The plane that passes through the point 共1, 2, 3兲 and contains
57. x ⫹ y ⫹ z 苷 1,
共⫺1, ⫺2, ⫺3兲
the line x 苷 3t, y 苷 1 ⫹ t, z 苷 2 ⫺ t
35. The plane that passes through the point 共6, 0, ⫺2兲 and contains
the line x 苷 4 ⫺ 2t, y 苷 3 ⫹ 5t, z 苷 7 ⫹ 4 t
36. The plane that passes through the point 共1, ⫺1, 1兲 and
contains the line with symmetric equations x 苷 2y 苷 3z
37. The plane that passes through the point 共⫺1, 2, 1兲 and contains
the line of intersection of the planes x ⫹ y ⫺ z 苷 2 and
2 x ⫺ y ⫹ 3z 苷 1
x ⫹ 2y ⫹ 2z 苷 1
58. 3x ⫺ 2y ⫹ z 苷 1,
2x ⫹ y ⫺ 3z 苷 3
59–60 Find symmetric equations for the line of intersection of the
planes.
59. 5x ⫺ 2y ⫺ 2z 苷 1,
60. z 苷 2x ⫺ y ⫺ 5,
4x ⫹ y ⫹ z 苷 6
z 苷 4x ⫹ 3y ⫺ 5
38. The plane that passes through the points 共0, ⫺2, 5兲 and
61. Find an equation for the plane consisting of all points that are
39. The plane that passes through the point 共1, 5, 1兲 and is perpen-
62. Find an equation for the plane consisting of all points that are
40. The plane that passes through the line of intersection of the
63. Find an equation of the plane with x-intercept a, y-intercept b,
共⫺1, 3, 1兲 and is perpendicular to the plane 2z 苷 5x ⫹ 4y
dicular to the planes 2x ⫹ y ⫺ 2z 苷 2 and x ⫹ 3z 苷 4
planes x ⫺ z 苷 1 and y ⫹ 2z 苷 3 and is perpendicular to the
plane x ⫹ y ⫺ 2z 苷 1
equidistant from the points 共1, 0, ⫺2兲 and 共3, 4, 0兲.
equidistant from the points 共2, 5, 5兲 and 共⫺6, 3, 1兲.
and z-intercept c.
64. (a) Find the point at which the given lines intersect:
41– 44 Use intercepts to help sketch the plane.
r 苷 具1, 1, 0典 ⫹ t 具1, ⫺1, 2典
41. 2x ⫹ 5y ⫹ z 苷 10
42. 3x ⫹ y ⫹ 2z 苷 6
r 苷 具2, 0, 2典 ⫹ s具⫺1, 1, 0典
43. 6x ⫺ 3y ⫹ 4z 苷 6
44. 6x ⫹ 5y ⫺ 3z 苷 15
45– 47 Find the point at which the line intersects the given plane.
45. x 苷 3 ⫺ t, y 苷 2 ⫹ t, z 苷 5t ;
x ⫺ y ⫹ 2z 苷 9
46. x 苷 1 ⫹ 2t, y 苷 4t, z 苷 2 ⫺ 3t ;
47. x 苷 y ⫺ 1 苷 2z ;
x ⫹ 2y ⫺ z ⫹ 1 苷 0
4x ⫺ y ⫹ 3z 苷 8
48. Where does the line through 共1, 0, 1兲 and 共4, ⫺2, 2兲 intersect
the plane x ⫹ y ⫹ z 苷 6 ?
49. Find direction numbers for the line of intersection of the planes
x ⫹ y ⫹ z 苷 1 and x ⫹ z 苷 0.
50. Find the cosine of the angle between the planes x ⫹ y ⫹ z 苷 0
and x ⫹ 2y ⫹ 3z 苷 1.
(b) Find an equation of the plane that contains these lines.
65. Find parametric equations for the line through the point
共0, 1, 2兲 that is parallel to the plane x ⫹ y ⫹ z 苷 2 and
perpendicular to the line x 苷 1 ⫹ t, y 苷 1 ⫺ t, z 苷 2t.
66. Find parametric equations for the line through the point
共0, 1, 2兲 that is perpendicular to the line x 苷 1 ⫹ t,
y 苷 1 ⫺ t, z 苷 2t and intersects this line.
67. Which of the following four planes are parallel? Are any of
them identical?
P1 : 3x ⫹ 6y ⫺ 3z 苷 6
P2 : 4x ⫺ 12y ⫹ 8z 苷 5
P3 : 9y 苷 1 ⫹ 3x ⫹ 6z
P4 : z 苷 x ⫹ 2y ⫺ 2
68. Which of the following four lines are parallel? Are any of them
identical?
51–56 Determine whether the planes are parallel, perpendicular, or
neither. If neither, find the angle between them.
L 1 : x 苷 1 ⫹ 6t, y 苷 1 ⫺ 3t, z 苷 12t ⫹ 5
51. x ⫹ 4y ⫺ 3z 苷 1,
L 3 : 2x ⫺ 2 苷 4 ⫺ 4y 苷 z ⫹ 1
52. 2z 苷 4y ⫺ x,
53. x ⫹ y ⫹ z 苷 1,
3x ⫺ 12y ⫹ 6z 苷 1
x⫺y⫹z苷1
54. 2 x ⫺ 3y ⫹ 4z 苷 5 ,
55. x 苷 4y ⫺ 2z,
L 2 : x 苷 1 ⫹ 2t, y 苷 t,
⫺3x ⫹ 6y ⫹ 7z 苷 0
x ⫹ 6y ⫹ 4z 苷 3
8y 苷 1 ⫹ 2 x ⫹ 4z
56. x ⫹ 2y ⫹ 2z 苷 1,
2x ⫺ y ⫹ 2z 苷 1
z 苷 1 ⫹ 4t
L 4 : r 苷 具3, 1, 5典 ⫹ t 具4, 2, 8典
69–70 Use the formula in Exercise 45 in Section 12.4 to find the
distance from the point to the given line.
69. 共4, 1, ⫺2兲;
70. 共0, 1, 3兲;
x 苷 1 ⫹ t, y 苷 3 ⫺ 2t, z 苷 4 ⫺ 3t
x 苷 2t, y 苷 6 ⫺ 2t, z 苷 3 ⫹ t
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VECTORS AND THE GEOMETRY OF SPACE
71–72 Find the distance from the point to the given plane.
71. 共1, ⫺2, 4兲,
3x ⫹ 2y ⫹ 6z 苷 5
72. 共⫺6, 3, 5兲,
x ⫺ 2y ⫺ 4z 苷 8
74. 6z 苷 4y ⫺ 2x,
4x ⫺ 6y ⫹ 2z 苷 3
75. Show that the distance between the parallel planes
ax ⫹ by ⫹ cz ⫹ d1 苷 0 and ax ⫹ by ⫹ cz ⫹ d2 苷 0 is
ⱍ
Let L 2 be the line through the points 共1, ⫺1, 1兲 and 共4, 1, 3兲.
Find the distance between L1 and L 2.
80. Let L1 be the line through the points 共1, 2, 6兲 and 共2, 4, 8兲.
Let L 2 be the line of intersection of the planes ␲1 and ␲ 2,
where ␲1 is the plane x ⫺ y ⫹ 2z ⫹ 1 苷 0 and ␲ 2 is the plane
through the points 共3, 2, ⫺1兲, 共0, 0, 1兲, and 共1, 2, 1兲. Calculate
the distance between L1 and L 2.
9z 苷 1 ⫺ 3x ⫹ 6y
D苷
equations x 苷 1 ⫹ t, y 苷 1 ⫹ 6t, z 苷 2t, and x 苷 1 ⫹ 2s,
y 苷 5 ⫹ 15s, z 苷 ⫺2 ⫹ 6s.
79. Let L1 be the line through the origin and the point 共2, 0, ⫺1兲.
73–74 Find the distance between the given parallel planes.
73. 2x ⫺ 3y ⫹ z 苷 4,
78. Find the distance between the skew lines with parametric
ⱍ
d1 ⫺ d2
sa 2 ⫹ b 2 ⫹ c 2
81. If a, b, and c are not all 0, show that the equation
ax ⫹ by ⫹ cz ⫹ d 苷 0 represents a plane and 具a, b, c典 is
a normal vector to the plane.
Hint: Suppose a 苷 0 and rewrite the equation in the form
冉 冊
a x⫹
76. Find equations of the planes that are parallel to the plane
x ⫹ 2y ⫺ 2z 苷 1 and two units away from it.
77. Show that the lines with symmetric equations x 苷 y 苷 z and
x ⫹ 1 苷 y兾2 苷 z兾3 are skew, and find the distance between
these lines.
d
a
⫹ b共 y ⫺ 0兲 ⫹ c共z ⫺ 0兲 苷 0
82. Give a geometric description of each family of planes.
(a) x ⫹ y ⫹ z 苷 c
(c) y cos ␪ ⫹ z sin ␪ 苷 1
(b) x ⫹ y ⫹ cz 苷 1
L A B O R AT O R Y P R O J E C T PUTTING 3D IN PERSPECTIVE
Computer graphics programmers face the same challenge as the great painters of the past: how
to represent a three-dimensional scene as a flat image on a two-dimensional plane (a screen or a
canvas). To create the illusion of perspective, in which closer objects appear larger than those
farther away, three-dimensional objects in the computer’s memory are projected onto a rectangular screen window from a viewpoint where the eye, or camera, is located. The viewing
volume––the portion of space that will be visible––is the region contained by the four planes that
pass through the viewpoint and an edge of the screen window. If objects in the scene extend
beyond these four planes, they must be truncated before pixel data are sent to the screen. These
planes are therefore called clipping planes.
1. Suppose the screen is represented by a rectangle in the yz-plane with vertices 共0, ⫾400, 0兲
and 共0, ⫾400, 600兲, and the camera is placed at 共1000, 0, 0兲. A line L in the scene passes
through the points 共230, ⫺285, 102兲 and 共860, 105, 264兲. At what points should L be clipped
by the clipping planes?
2. If the clipped line segment is projected on the screen window, identify the resulting line
segment.
3. Use parametric equations to plot the edges of the screen window, the clipped line segment,
and its projection on the screen window. Then add sight lines connecting the viewpoint to
each end of the clipped segments to verify that the projection is correct.
4. A rectangle with vertices 共621, ⫺147, 206兲, 共563, 31, 242兲, 共657, ⫺111, 86兲, and
共599, 67, 122兲 is added to the scene. The line L intersects this rectangle. To make the rectangle appear opaque, a programmer can use hidden line rendering, which removes portions
of objects that are behind other objects. Identify the portion of L that should be removed.
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VECTORS AND THE GEOMETRY OF SPACE
Applications of Quadric Surfaces
© David Frazier / Corbis
© Mark C. Burnett / Photo Researchers, Inc
Examples of quadric surfaces can be found in the world around us. In fact, the world itself
is a good example. Although the earth is commonly modeled as a sphere, a more accurate
model is an ellipsoid because the earth’s rotation has caused a flattening at the poles. (See
Exercise 47.)
Circular paraboloids, obtained by rotating a parabola about its axis, are used to collect
and reflect light, sound, and radio and television signals. In a radio telescope, for instance,
signals from distant stars that strike the bowl are all reflected to the receiver at the focus and
are therefore amplified. (The idea is explained in Problem 16 on page 196.) The same principle applies to microphones and satellite dishes in the shape of paraboloids.
Cooling towers for nuclear reactors are usually designed in the shape of hyperboloids of
one sheet for reasons of structural stability. Pairs of hyperboloids are used to transmit rotational motion between skew axes. (The cogs of the gears are the generating lines of the
hyperboloids. See Exercise 49.)
A satellite dish reflects signals to
the focus of a paraboloid.
12.6
Nuclear reactors have cooling towers
in the shape of hyperboloids.
Exercises
1. (a) What does the equation y 苷 x 2 represent as a curve in ⺢ 2 ?
(b) What does it represent as a surface in ⺢ ?
(c) What does the equation z 苷 y 2 represent?
3
(b) Sketch the graph of y 苷 e as a surface in ⺢ .
(c) Describe and sketch the surface z 苷 e y.
x
3
3–8 Describe and sketch the surface.
3. y 2 ⫹ 4z 2 苷 4
4. z 苷 4 ⫺ x 2
Graphing calculator or computer required
5. z 苷 1 ⫺ y 2
6. y 苷 z 2
7. xy 苷 1
8. z 苷 sin y
9. (a) Find and identify the traces of the quadric surface
2. (a) Sketch the graph of y 苷 e x as a curve in ⺢ 2.
;
Hyperboloids produce gear transmission.
x 2 ⫹ y 2 ⫺ z 2 苷 1 and explain why the graph looks like
the graph of the hyperboloid of one sheet in Table 1.
(b) If we change the equation in part (a) to x 2 ⫺ y 2 ⫹ z 2 苷 1,
how is the graph affected?
(c) What if we change the equation in part (a) to
x 2 ⫹ y 2 ⫹ 2y ⫺ z 2 苷 0?
1. Homework Hints available at stewartcalculus.com
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SECTION 12.6
10. (a) Find and identify the traces of the quadric surface
CYLINDERS AND QUADRIC SURFACES
857
29–36 Reduce the equation to one of the standard forms, classify
the surface, and sketch it.
⫺x 2 ⫺ y 2 ⫹ z 2 苷 1 and explain why the graph looks like
the graph of the hyperboloid of two sheets in Table 1.
(b) If the equation in part (a) is changed to x 2 ⫺ y 2 ⫺ z 2 苷 1,
what happens to the graph? Sketch the new graph.
29. y 2 苷 x 2 ⫹ 9 z 2
30. 4x 2 ⫺ y ⫹ 2z 2 苷 0
31. x 2 ⫹ 2y ⫺ 2z 2 苷 0
32. y 2 苷 x 2 ⫹ 4z 2 ⫹ 4
1
11–20 Use traces to sketch and identify the surface.
33. 4x 2 ⫹ y 2 ⫹ 4 z 2 ⫺ 4y ⫺ 24z ⫹ 36 苷 0
11. x 苷 y 2 ⫹ 4z 2
12. 9x 2 ⫺ y 2 ⫹ z 2 苷 0
34. 4y 2 ⫹ z 2 ⫺ x ⫺ 16y ⫺ 4z ⫹ 20 苷 0
13. x 2 苷 y 2 ⫹ 4z 2
14. 25x 2 ⫹ 4y 2 ⫹ z 2 苷 100
35. x 2 ⫺ y 2 ⫹ z 2 ⫺ 4x ⫺ 2y ⫺ 2z ⫹ 4 苷 0
15. ⫺x 2 ⫹ 4y 2 ⫺ z 2 苷 4
16. 4x 2 ⫹ 9y 2 ⫹ z 苷 0
36. x 2 ⫺ y 2 ⫹ z 2 ⫺ 2x ⫹ 2y ⫹ 4z ⫹ 2 苷 0
17. 36x 2 ⫹ y 2 ⫹ 36z 2 苷 36
18. 4x 2 ⫺ 16y 2 ⫹ z 2 苷 16
19. y 苷 z 2 ⫺ x 2
20. x 苷 y 2 ⫺ z 2
; 37– 40 Use a computer with three-dimensional graphing software to
graph the surface. Experiment with viewpoints and with domains
for the variables until you get a good view of the surface.
21–28 Match the equation with its graph (labeled I–VIII). Give
reasons for your choice.
37. ⫺4x 2 ⫺ y 2 ⫹ z 2 苷 1
38. x 2 ⫺ y 2 ⫺ z 苷 0
21. x 2 ⫹ 4y 2 ⫹ 9z 2 苷 1
22. 9x 2 ⫹ 4y 2 ⫹ z 2 苷 1
39. ⫺4x 2 ⫺ y 2 ⫹ z 2 苷 0
40. x 2 ⫺ 6x ⫹ 4y 2 ⫺ z 苷 0
23. x 2 ⫺ y 2 ⫹ z 2 苷 1
24. ⫺x 2 ⫹ y 2 ⫺ z 2 苷 1
25. y 苷 2x 2 ⫹ z 2
26. y 2 苷 x 2 ⫹ 2z 2
27. x ⫹ 2z 苷 1
28. y 苷 x ⫺ z
2
2
2
z
I
41. Sketch the region bounded by the surfaces z 苷 sx 2 ⫹ y 2
and x 2 ⫹ y 2 苷 1 for 1 艋 z 艋 2.
2
42. Sketch the region bounded by the paraboloids z 苷 x 2 ⫹ y 2
and z 苷 2 ⫺ x 2 ⫺ y 2.
z
II
43. Find an equation for the surface obtained by rotating the
parabola y 苷 x 2 about the y-axis.
y
x
y
x
44. Find an equation for the surface obtained by rotating the line
x 苷 3y about the x-axis.
z
III
45. Find an equation for the surface consisting of all points that
z
IV
are equidistant from the point 共⫺1, 0, 0兲 and the plane x 苷 1.
Identify the surface.
46. Find an equation for the surface consisting of all points P for
y
which the distance from P to the x-axis is twice the distance
from P to the yz-plane. Identify the surface.
y
x
x
z
V
y
x
z
VII
y
x
VIII
y
x
47. Traditionally, the earth’s surface has been modeled as a sphere,
z
VI
z
but the World Geodetic System of 1984 (WGS-84) uses an
ellipsoid as a more accurate model. It places the center of the
earth at the origin and the north pole on the positive z-axis.
The distance from the center to the poles is 6356.523 km and
the distance to a point on the equator is 6378.137 km.
(a) Find an equation of the earth’s surface as used by
WGS-84.
(b) Curves of equal latitude are traces in the planes z 苷 k.
What is the shape of these curves?
(c) Meridians (curves of equal longitude) are traces in
planes of the form y 苷 mx. What is the shape of these
meridians?
48. A cooling tower for a nuclear reactor is to be constructed in
the shape of a hyperboloid of one sheet (see the photo on
page 856). The diameter at the base is 280 m and the minimum
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VECTORS AND THE GEOMETRY OF SPACE
diameter, 500 m above the base, is 200 m. Find an equation
for the tower.
49. Show that if the point 共a, b, c兲 lies on the hyperbolic parabo-
loid z 苷 y 2 ⫺ x 2, then the lines with parametric equations
x 苷 a ⫹ t, y 苷 b ⫹ t, z 苷 c ⫹ 2共b ⫺ a兲t and x 苷 a ⫹ t,
y 苷 b ⫺ t, z 苷 c ⫺ 2共b ⫹ a兲t both lie entirely on this paraboloid. (This shows that the hyperbolic paraboloid is what is
called a ruled surface; that is, it can be generated by the
motion of a straight line. In fact, this exercise shows that
through each point on the hyperbolic paraboloid there are two
12
generating lines. The only other quadric surfaces that are ruled
surfaces are cylinders, cones, and hyperboloids of one sheet.)
50. Show that the curve of intersection of the surfaces
x 2 ⫹ 2y 2 ⫺ z 2 ⫹ 3x 苷 1 and 2x 2 ⫹ 4y 2 ⫺ 2z 2 ⫺ 5y 苷 0
lies in a plane.
2
2
2
; 51. Graph the surfaces z 苷 x ⫹ y and z 苷 1 ⫺ y on a common
ⱍ ⱍ
ⱍ ⱍ
screen using the domain x 艋 1.2, y 艋 1.2 and observe the
curve of intersection of these surfaces. Show that the projection
of this curve onto the xy-plane is an ellipse.
Review
Concept Check
1. What is the difference between a vector and a scalar?
11. How do you find a vector perpendicular to a plane?
2. How do you add two vectors geometrically? How do you add
12. How do you find the angle between two intersecting planes?
them algebraically?
3. If a is a vector and c is a scalar, how is ca related to a
geometrically? How do you find ca algebraically?
13. Write a vector equation, parametric equations, and symmetric
equations for a line.
4. How do you find the vector from one point to another?
14. Write a vector equation and a scalar equation for a plane.
5. How do you find the dot product a ⴢ b of two vectors if you
15. (a) How do you tell if two vectors are parallel?
know their lengths and the angle between them? What if you
know their components?
6. How are dot products useful?
7. Write expressions for the scalar and vector projections of b
onto a. Illustrate with diagrams.
8. How do you find the cross product a ⫻ b of two vectors if you
know their lengths and the angle between them? What if you
know their components?
9. How are cross products useful?
10. (a) How do you find the area of the parallelogram determined
by a and b?
(b) How do you find the volume of the parallelepiped
determined by a, b, and c?
(b) How do you tell if two vectors are perpendicular?
(c) How do you tell if two planes are parallel?
16. (a) Describe a method for determining whether three points
P, Q, and R lie on the same line.
(b) Describe a method for determining whether four points
P, Q, R, and S lie in the same plane.
17. (a) How do you find the distance from a point to a line?
(b) How do you find the distance from a point to a plane?
(c) How do you find the distance between two lines?
18. What are the traces of a surface? How do you find them?
19. Write equations in standard form of the six types of quadric
surfaces.
True-False Quiz
Determine whether the statement is true or false. If it is true, explain why.
If it is false, explain why or give an example that disproves the statement.
1. If u 苷 具u1, u2 典 and v 苷 具 v1, v2 典 , then u ⴢ v 苷 具u1v1, u2 v2 典 .
ⱍ
ⱍ ⱍ ⱍ ⱍ ⱍ
For any vectors u and v in V , ⱍ u ⴢ v ⱍ 苷 ⱍ u ⱍⱍ v ⱍ.
For any vectors u and v in V , ⱍ u ⫻ v ⱍ 苷 ⱍ u ⱍⱍ v ⱍ.
2. For any vectors u and v in V3 , u ⫹ v 苷 u ⫹ v .
3.
4.
3
3
5. For any vectors u and v in V3 , u ⴢ v 苷 v ⴢ u.
6. For any vectors u and v in V3 , u ⫻ v 苷 v ⫻ u.
ⱍ
ⱍ ⱍ
ⱍ
7. For any vectors u and v in V3 , u ⫻ v 苷 v ⫻ u .
8. For any vectors u and v in V3 and any scalar k,
k共u ⴢ v兲 苷 共k u兲 ⴢ v.
9. For any vectors u and v in V3 and any scalar k,
k共u ⫻ v兲 苷 共k u兲 ⫻ v.
10. For any vectors u, v, and w in V3,
共u ⫹ v兲 ⫻ w 苷 u ⫻ w ⫹ v ⫻ w.
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REVIEW
859
16. A linear equation Ax ⫹ By ⫹ Cz ⫹ D 苷 0 represents a line
11. For any vectors u, v, and w in V3,
u ⴢ 共v ⫻ w兲 苷 共u ⫻ v兲 ⴢ w.
in space.
17. The set of points {共x, y, z兲
12. For any vectors u, v, and w in V3 ,
u ⫻ 共v ⫻ w兲 苷 共u ⫻ v兲 ⫻ w.
ⱍx
2
⫹ y 2 苷 1} is a circle.
18. In ⺢ 3 the graph of y 苷 x 2 is a paraboloid.
13. For any vectors u and v in V3 , 共u ⫻ v兲 ⴢ u 苷 0.
19. If u ⴢ v 苷 0 , then u 苷 0 or v 苷 0.
14. For any vectors u and v in V3 , 共u ⫹ v兲 ⫻ v 苷 u ⫻ v.
20. If u ⫻ v 苷 0, then u 苷 0 or v 苷 0.
15. The vector 具3, ⫺1, 2 典 is parallel to the plane
6x ⫺ 2y ⫹ 4z 苷 1.
21. If u ⴢ v 苷 0 and u ⫻ v 苷 0, then u 苷 0 or v 苷 0.
ⱍ
ⱍ ⱍ ⱍⱍ v ⱍ.
22. If u and v are in V3 , then u ⴢ v 艋 u
Exercises
1. (a) Find an equation of the sphere that passes through the point
共6, ⫺2, 3兲 and has center 共⫺1, 2, 1兲.
(b) Find the curve in which this sphere intersects the yz-plane.
(c) Find the center and radius of the sphere
x 2 ⫹ y 2 ⫹ z 2 ⫺ 8x ⫹ 2y ⫹ 6z ⫹ 1 苷 0
2. Copy the vectors in the figure and use them to draw each of the
following vectors.
(a) a ⫹ b
(b) a ⫺ b
(d) 2 a ⫹ b
(c) ⫺ a
1
2
6. Find two unit vectors that are orthogonal to both j ⫹ 2 k
and i ⫺ 2 j ⫹ 3 k.
7. Suppose that u ⴢ 共v ⫻ w兲 苷 2. Find
(a) 共u ⫻ v兲 ⴢ w
(c) v ⴢ 共u ⫻ w兲
(b) u ⴢ 共w ⫻ v兲
(d) 共u ⫻ v兲 ⴢ v
8. Show that if a, b, and c are in V3 , then
共a ⫻ b兲 ⴢ 关共b ⫻ c兲 ⫻ 共c ⫻ a兲兴 苷 关a ⴢ 共b ⫻ c兲兴 2
9. Find the acute angle between two diagonals of a cube.
10. Given the points A共1, 0, 1兲, B共2, 3, 0兲, C共⫺1, 1, 4兲, and
D共0, 3, 2兲, find the volume of the parallelepiped with adjacent
edges AB, AC, and AD.
a
b
11. (a) Find a vector perpendicular to the plane through the points
A共1, 0, 0兲, B共2, 0, ⫺1兲, and C共1, 4, 3兲.
(b) Find the area of triangle ABC.
3. If u and v are the vectors shown in the figure, find u ⴢ v and
ⱍ u ⫻ v ⱍ. Is u ⫻ v directed into the page or out of it?
12. A constant force F 苷 3 i ⫹ 5 j ⫹ 10 k moves an object along
the line segment from 共1, 0, 2兲 to 共5, 3, 8兲. Find the work done
if the distance is measured in meters and the force in newtons.
13. A boat is pulled onto shore using two ropes, as shown in the
diagram. If a force of 255 N is needed, find the magnitude of
the force in each rope.
|v|=3
45°
|u|=2
20° 255 N
30°
4. Calculate the given quantity if
a 苷 i ⫹ j ⫺ 2k
b 苷 3i ⫺ 2j ⫹ k
c 苷 j ⫺ 5k
(a)
(c)
(e)
(g)
(i)
(k)
14. Find the magnitude of the torque about P if a 50-N force is
ⱍ ⱍ
(b) b
2a ⫹ 3b
(d) a ⫻ b
aⴢb
(f ) a ⴢ 共b ⫻ c兲
b⫻c
(h) a ⫻ 共b ⫻ c兲
c⫻c
( j) proj a b
comp a b
The angle between a and b (correct to the nearest degree)
ⱍ
applied as shown.
50 N
ⱍ
30°
40 cm
5. Find the values of x such that the vectors 具3, 2, x典 and
具2x, 4, x典 are orthogonal.
P
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860
CHAPTER 12
VECTORS AND THE GEOMETRY OF SPACE
15–17 Find parametric equations for the line.
15. The line through 共4, ⫺1, 2兲 and 共1, 1, 5兲
16. The line through 共1, 0, ⫺1兲 and parallel to the line
1
3
共x ⫺ 4兲 苷 y 苷 z ⫹ 2
1
2
17. The line through 共⫺2, 2, 4兲 and perpendicular to the
plane 2x ⫺ y ⫹ 5z 苷 12
18–20 Find an equation of the plane.
18. The plane through 共2, 1, 0兲 and parallel to x ⫹ 4y ⫺ 3z 苷 1
19. The plane through 共3, ⫺1, 1兲, 共4, 0, 2兲, and 共6, 3, 1兲
20. The plane through 共1, 2, ⫺2兲 that contains the line
x 苷 2t, y 苷 3 ⫺ t, z 苷 1 ⫹ 3t
21. Find the point in which the line with parametric equations
x 苷 2 ⫺ t, y 苷 1 ⫹ 3t, z 苷 4t intersects the plane
2 x ⫺ y ⫹ z 苷 2.
22. Find the distance from the origin to the line
x 苷 1 ⫹ t, y 苷 2 ⫺ t, z 苷 ⫺1 ⫹ 2t.
23. Determine whether the lines given by the symmetric
equations
x⫺1
y⫺2
z⫺3
苷
苷
2
3
4
and
x⫹1
y⫺3
z⫹5
苷
苷
6
⫺1
2
are parallel, skew, or intersecting.
24. (a) Show that the planes x ⫹ y ⫺ z 苷 1 and
2x ⫺ 3y ⫹ 4z 苷 5 are neither parallel nor perpendicular.
(b) Find, correct to the nearest degree, the angle between these
planes.
25. Find an equation of the plane through the line of intersection of
the planes x ⫺ z 苷 1 and y ⫹ 2z 苷 3 and perpendicular to the
plane x ⫹ y ⫺ 2z 苷 1.
26. (a) Find an equation of the plane that passes through the points
A共2, 1, 1兲, B共⫺1, ⫺1, 10兲, and C共1, 3, ⫺4兲.
(b) Find symmetric equations for the line through B that is
perpendicular to the plane in part (a).
(c) A second plane passes through 共2, 0, 4兲 and has normal
vector 具2, ⫺4, ⫺3 典 . Show that the acute angle between the
planes is approximately 43⬚.
(d) Find parametric equations for the line of intersection of the
two planes.
27. Find the distance between the planes 3x ⫹ y ⫺ 4z 苷 2
and 3x ⫹ y ⫺ 4z 苷 24.
28–36 Identify and sketch the graph of each surface.
28. x 苷 3
29. x 苷 z
30. y 苷 z
31. x 2 苷 y 2 ⫹ 4z 2
2
32. 4x ⫺ y ⫹ 2z 苷 4
33. ⫺4x 2 ⫹ y 2 ⫺ 4z 2 苷 4
34. y 2 ⫹ z 2 苷 1 ⫹ x 2
35. 4x 2 ⫹ 4y 2 ⫺ 8y ⫹ z 2 苷 0
36. x 苷 y 2 ⫹ z 2 ⫺ 2y ⫺ 4z ⫹ 5
37. An ellipsoid is created by rotating the ellipse 4x 2 ⫹ y 2 苷 16
about the x-axis. Find an equation of the ellipsoid.
38. A surface consists of all points P such that the distance from P
to the plane y 苷 1 is twice the distance from P to the point
共0, ⫺1, 0兲. Find an equation for this surface and identify it.
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Problems Plus
1. Each edge of a cubical box has length 1 m. The box contains nine spherical balls with the
1m
same radius r. The center of one ball is at the center of the cube and it touches the other eight
balls. Each of the other eight balls touches three sides of the box. Thus the balls are tightly
packed in the box. (See the figure.) Find r. (If you have trouble with this problem, read about
the problem-solving strategy entitled Use Analogy on page 97.)
2. Let B be a solid box with length L , width W, and height H. Let S be the set of all points that
1m
are a distance at most 1 from some point of B. Express the volume of S in terms of L , W,
and H.
1m
FIGURE FOR PROBLEM 1
3. Let L be the line of intersection of the planes cx ⫹ y ⫹ z 苷 c and x ⫺ cy ⫹ cz 苷 ⫺1,
where c is a real number.
(a) Find symmetric equations for L .
(b) As the number c varies, the line L sweeps out a surface S. Find an equation for the curve
of intersection of S with the horizontal plane z 苷 t (the trace of S in the plane z 苷 t).
(c) Find the volume of the solid bounded by S and the planes z 苷 0 and z 苷 1.
4. A plane is capable of flying at a speed of 180 km兾h in still air. The pilot takes off from an
airfield and heads due north according to the plane’s compass. After 30 minutes of flight time,
the pilot notices that, due to the wind, the plane has actually traveled 80 km at an angle 5°
east of north.
(a) What is the wind velocity?
(b) In what direction should the pilot have headed to reach the intended destination?
ⱍ ⱍ
ⱍ ⱍ
5. Suppose v1 and v2 are vectors with v1 苷 2, v2 苷 3, and v1 ⴢ v2 苷 5. Let v3 苷 proj v v2,
1
v4 苷 projv v3, v5 苷 projv v4, and so on. Compute 冘⬁n苷1 vn .
2
3
ⱍ ⱍ
6. Find an equation of the largest sphere that passes through the point 共⫺1, 1, 4兲 and is such that
each of the points 共x, y, z兲 inside the sphere satisfies the condition
x 2 ⫹ y 2 ⫹ z 2 ⬍ 136 ⫹ 2共x ⫹ 2y ⫹ 3z兲
N
F
7. Suppose a block of mass m is placed on an inclined plane, as shown in the figure. The block’s
descent down the plane is slowed by friction; if ␪ is not too large, friction will prevent the
block from moving at all. The forces acting on the block are the weight W, where W 苷 mt
( t is the acceleration due to gravity); the normal force N (the normal component of the reactionary force of the plane on the block), where N 苷 n; and the force F due to friction,
which acts parallel to the inclined plane, opposing the direction of motion. If the block is at
rest and ␪ is increased, F must also increase until ultimately F reaches its maximum,
beyond which the block begins to slide. At this angle ␪s , it has been observed that F is
proportional to n. Thus, when F is maximal, we can say that F 苷 ␮ s n, where ␮ s is
called the coefficient of static friction and depends on the materials that are in contact.
(a) Observe that N ⫹ F ⫹ W ⫽ 0 and deduce that ␮ s 苷 tan共␪s兲 .
(b) Suppose that, for ␪ ⬎ ␪ s , an additional outside force H is applied to the block, horizontally from the left, and let H 苷 h. If h is small, the block may still slide down the
plane; if h is large enough, the block will move up the plane. Let h min be the smallest
value of h that allows the block to remain motionless (so that F is maximal).
By choosing the coordinate axes so that F lies along the x-axis, resolve each force into
components parallel and perpendicular to the inclined plane and show that
ⱍ ⱍ
W
¨
FIGURE FOR PROBLEM 7
ⱍ ⱍ
ⱍ ⱍ
ⱍ ⱍ
ⱍ ⱍ
ⱍ ⱍ
ⱍ ⱍ
ⱍ ⱍ
ⱍ ⱍ
h min sin ␪ ⫹ mt cos ␪ 苷 n
(c) Show that
and
h min cos ␪ ⫹ ␮ s n 苷 mt sin ␪
h min 苷 mt tan共␪ ⫺ ␪s 兲
Does this equation seem reasonable? Does it make sense for ␪ 苷 ␪s ? As ␪ l 90⬚ ?
Explain.
861
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(d) Let h max be the largest value of h that allows the block to remain motionless. (In which
direction is F heading?) Show that
h max 苷 mt tan共␪ ⫹ ␪s 兲
Does this equation seem reasonable? Explain.
8. A solid has the following properties. When illuminated by rays parallel to the z-axis, its
shadow is a circular disk. If the rays are parallel to the y-axis, its shadow is a square. If the
rays are parallel to the x-axis, its shadow is an isosceles triangle. (In Exercise 44 in Section 12.1 you were asked to describe and sketch an example of such a solid, but there are
many such solids.) Assume that the projection onto the xz-plane is a square whose sides have
length 1.
(a) What is the volume of the largest such solid?
(b) Is there a smallest volume?
862