# Comparing this equation with the standard form, we see that it is the equation of a

## Full text

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814 CHAPTER 12 VECTORS AND THE GEOMETRY OF SPACE

1. Homework Hints available at stewartcalculus.com

1. Suppose you start at the origin, move along the -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 , , , and

on a single set of coordinate axes.

3. Which of the points , , and

is closest to the -plane? Which point lies in the -plane?

4. What are the projections of the point (2, 3, 5) on the -, -, and -planes? Draw a rectangular box with the origin and

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 represented by the equa- tion .

6. (a) What does the equation represent in ? What does it represent in ? Illustrate with sketches.

(b) What does the equation represent in ? What does represent? What does the pair of equations , represent? In other words, describe the set of points

such that and . Illustrate with a sketch.

7–8 Find the lengths of the sides of the triangle . Is it a right triangle? Is it an isosceles triangle?

7. , ,

8. , ,

x

A共⫺4, 0, ⫺1兲

yz xz

B共3, 1, ⫺5兲 C共2, 4, 6兲

xy yz xz

3 x⫹ y 苷 2

x苷 4 ⺢2

3

y苷 3 ⺢3

z苷 5 y苷 3

z苷 5

PQR

P共3, ⫺2, ⫺3兲 Q共7, 0, 1兲 R共1, 2, 1兲 P共2, ⫺1, 0兲 Q共4, 1, 1兲 R共4, ⫺5, 4兲

9. Determine whether the points lie on straight line.

(a) , ,

(b) , ,

10. Find the distance from to each of the following.

(a) The -plane (b) The -plane (c) The -plane (d) The -axis (e) The -axis (f ) The -axis

11. Find an equation of the sphere with center and radius 5. What is the intersection of this sphere with the

-plane?

12. Find an equation of the sphere with center and radius 5. Describe its intersection with each of the coordinate planes.

13. Find an equation of the sphere that passes through the point and has center .

14. Find an equation of the sphere that passes through the origin and whose center is .

15–18 Show that the equation represents a sphere, and find its center and radius.

15.

16.

17.

18.

D共0, ⫺5, 5兲 E共1, ⫺2, 4兲 F共3, 4, 2兲 C共1, 3, 3兲 B共3, 7, ⫺2兲

A共2, 4, 2兲

x2⫹ y2⫹ z2⫺ 2x ⫺ 4y ⫹ 8z 苷 15 x2⫹ y2⫹ z2⫹ 8x ⫺ 6y ⫹ 2z ⫹ 17 苷 0 2x2⫹ 2y2⫹ 2z2苷 8x ⫺ 24z ⫹ 1 3x2⫹ 3y2⫹ 3z2苷 10 ⫹ 6y ⫹ 12z

xy yz

xz x

y z

SOLUTION

3

2

2

2

z

2

2

2

2

2

2

z

x2

2

2

x2

2

2

### 苷 4

EXAMPLE 7

FIGURE 13 0 1 2

z

x y

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SECTION 12.2 VECTORS 815 19. (a) Prove that the midpoint of the line segment from

to is

(b) Find the lengths of the medians of the triangle with ver tices

, , and .

20. Find an equation of a sphere if one of its diameters has end - points and .

21. Find equations of the spheres with center that touch (a) the -plane, (b) the -plane, (c) the -plane.

22. Find an equation of the largest sphere with center (5, 4, 9) that is contained in the first octant.

23–34 Describe in words the region of represented by the equa- tions or inequalities.

23. 24.

25. 26.

27. 28.

29. , 30.

31. 32.

33. 34.

35–38 Write inequalities to describe the region.

35. The region between the -plane and the vertical plane 36. The solid cylinder that lies on or below the plane and on

or above the disk in the -plane with center the origin and radius 2

37. The region consisting of all points between (but not on) the spheres of radius and centered at the origin, where 38. The solid upper hemisphere of the sphere of radius 2 centered

at the origin

39. The figure shows a line in space and a second line

which is the projection of on the -plane. (In other words,

## 冉

x1⫹ x2 2, y1⫹ y2 2, z1⫹ z2 2

## 冊

A共1, 2, 3兲 B共⫺2, 0, 5兲 C共4, 1, 5兲 共2, 1, 4兲 共4, 3, 10兲

xy yz xz

3

x苷 5 y苷 ⫺2

y⬍ 8 x艌 ⫺3

0艋 z 艋 6 z2苷 1

x2⫹ y2苷 4 z 苷 ⫺1 y2⫹ z2苷 16 x2⫹ y2⫹ z2艋 3 x苷 z

x2⫹ z2艋 9 x2⫹ y2⫹ z2⬎ 2z

yz x苷 5

z苷 8 xy

r R r⬍ R

L1 L2,

L1 xy P2共x2, y2, z2P1共x1, y1, z1

the points on are directly beneath, or above, the points on .)

(a) Find the coordinates of the point on the line . (b) Locate on the diagram the points , , and , where

the line intersects the -plane, the -plane, and the -plane, respectively.

40. Consider the points such that the distance from to is twice the distance from to . 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 and . Describe the set.

42. Find the volume of the solid that lies inside both of the spheres

and

43. Find the distance between the spheres and .

44. Describe and sketch a solid with the following properties.

When illuminated by rays parallel to the -axis, its shadow is a circular disk. If the rays are parallel to the -axis, its shadow is a square. If the rays are parallel to the -axis, its shadow is an isosceles triangle.

L2

L1

P L1

A B C

L1 xy yz

xz

P P

A共⫺1, 5, 3兲 P B共6, 2, ⫺2兲

A共⫺1, 5, 3兲 B共6, 2, ⫺2兲

x2⫹ y2⫹ z2⫹ 4x ⫺ 2y ⫹ 4z ⫹ 5 苷 0 x2⫹ y2⫹ z2苷 4

x2⫹ y2⫹ z2苷 4 x2⫹ y2⫹ z2苷 4x ⫹ 4y ⫹ 4z ⫺ 11

z y x x

0 z

y 1

1 1

L™

P

AB

vl

A B

v A

v

B

### 12.2Vectors

FIGURE 1 Equivalent vectors A

B v

C

D u

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822 CHAPTER 12 VECTORS AND THE GEOMETRY OF SPACE

1. Homework Hints available at stewartcalculus.com

1. Are the following quantities vectors or scalars? Explain.

(a) The cost of a theater ticket (b) The current in a river

(c) The initial flight path from Houston to Dallas (d) The population of the world

2. What is the relationship between the point (4, 7) and the vector ? Illustrate with a sketch.

3. Name all the equal vectors in the parallelogram shown.

4. Write each combination of vectors as a single vector.

(a) PQl QRl

(b) RPl PSl

QSl

PSl

(d) RSl SPl

PQl 具4, 7典

B E

A

D C

###   

Q

R S

P

5. Copy the vectors in the figure and use them to draw the following vectors.

(a) (b)

(c) (d)

(e) (f )

6. Copy the vectors in the figure and use them to draw the following vectors.

(a) (b)

(c) (d)

(e) (f )

7. In the figure, the tip of and the tail of are both the midpoint of . Express and in terms of and .

u v u w

v w u v

v w u

v u  w u w  v

a b a b

1

2a 3b

a 2b 2b a

b a

c d

QR c d a b

b

a c

d P

Q

R

T1

T2

T1

T2

T2

T1

T1

T1

T2

T1

T1

T2

### ⬇ 539 i  337 j

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SECTION 12.2 VECTORS 823 8. If the vectors in the figure satisfy and

, what is ?

9–14 Find a vector with representation given by the directed line segment ABl

. Draw ABl

and the equivalent representation starting at the origin.

9. , 10. ,

11. , 12. ,

13. , 14. ,

15–18 Find the sum of the given vectors and illustrate geometrically.

15. , 16. ,

17. , 18. ,

19–22 Find a  b, 2a  3b, , and .

19. ,

20. ,

21. ,

22. ,

23–25 Find a unit vector that has the same direction as the given vector.

23. 24.

25.

26. Find a vector that has the same direction as but has length 6.

27–28 What is the angle between the given vector and the positive direction of the -axis?

27. 28.

29. If lies in the first quadrant and makes an angle with the positive -axis and , find 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 above the horizontal, find the horizontal and vertical components of the force.

31. A quarterback throws a football with angle of elevation and speed . Find the horizontal and vertical components of the velocity vector.

u

v

u v  w 苷 0

w

### ⱍ

u

v

w

a

A共1, 1兲 B共3, 2兲 A共4, 1兲 B共1, 2兲 B共0, 6兲 A共2, 1兲

B共2, 2兲 A共1, 3兲

B共4, 2, 1兲 A共4, 0, 2兲

B共2, 3, 1兲 A共0, 3, 1兲

a b

a

### ⱍ

b苷 具3, 6典 a苷 具5, 12典

b苷 i  2 j a苷 4 i  j

b苷 2 i  j  5k a苷 i  2 j  3k

b苷 2 j  k a苷 2 i  4 j  4 k

x

i s3 j 8 i 6j

兾3 v

v

### ⱍ

38 

40  60 ft兾s

3i  7j

32–33 Find the magnitude of the resultant force and the angle it makes with the positive -axis.

32. 33.

34. The magnitude of a velocity vector is called speed. Suppose that a wind is blowing from the direction N W at a speed of 50 km兾h. (This means that the direction from which the wind blows is west of the northerly direction.) A pilot is steering a plane in the direction N 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 resul tant 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 and with the horizontal. Find the tension in each wire and the magnitude of each tension.

37. A clothesline is tied between two poles, 8 m apart. The line 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 mid point 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?

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 and the speed of his boat is .

(a) In what direction should he steer?

(b) How long will the trip take?

x

300 N

200 N

60° 0 y

x

45 45

60

52 40

3 m 5 m

52° 40°

37° 37°

3.5 km兾h 13 km兾h

20 N

16 N 45° 0 y

x 30°

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824 CHAPTER 12 VECTORS AND THE GEOMETRY OF SPACE 40. Three forces act on an object. Two of the forces are at an angle

of 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 at the point .

42. (a) Find the unit vectors that are parallel to the tangent line to the curve at the point .

(b) Find the unit vectors that are perpendicular to the tangent line.

(c) Sketch the curve and the vectors in parts (a) and (b), all starting at .

43. If , , and are the vertices of a triangle, find ABl

 BCl

 CAl .

44. Let be the point on the line segment that is twice as far from as it is from . If OAl

, OBl

, and OCl , show that .

45. (a) Draw the vectors , , and (b) Show, by means of a sketch, that there are scalars and

such that .

(c) Use the sketch to estimate the values of and . (d) Find the exact values of and .

46. Suppose that and are nonzero vectors that are not parallel and is any vector in the plane determined by and . Give a geometric argument to show that can be written as

for suitable scalars and Then give an argu- ment using components.

47. If and , describe the set of all

points such that .

100

y苷 2 sin x 共兾6, 1兲 C

B A

AB C

ba

A B

c23a13b

b苷 具2, 1典 a苷 具3, 2典

c苷 具7, 1典.

t s c苷 sa  tb

t s t

s b a

b a c

c t.

s c苷 sa  tb

r0苷 具x0, y0, z0r苷 具x, y, z典

r r0

### ⱍ

c

48. If , , and , describe the

set of all points such that ,

where .

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 .

50. Prove Property 5 of vectors algebraically for the case . 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 first strikes the

-plane, as shown in the figure. Use the fact that the angle of incidence equals the angle of reflection to show that the direc- tion of the reflected ray is given by . 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.)

k

r1 r2

### ⱍ

n苷 2

n苷 3

a苷 具a1, a2, a3xz

b苷 具a1, a2, a3

b a

z

x

y r2苷 具x2, y2r1苷 具x1, y1

r苷 具x, y典

r r1



r r2

Deﬁnition

a

1

2

3

b

1

2

3

a

b a

a

1b1

2b2

3b3

a b

1

2

1

2

1b1

2b2

1

### 12.3The Dot Product

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830 CHAPTER 12 VECTORS AND THE GEOMETRY OF SPACE

1. Which of the following expressions are meaningful? Which are meaningless? Explain.

(a) (b)

(c) (d)

(e) (f )

2–10 Find .

2. ,

3. ,

4. ,

5. ,

6. ,

7. ,

8. ,

9. , , the angle between and is 10. , , the angle between and is

11–12 If u is a unit vector, find and .

11. 12.

13. (a) Show that .

(b) Show that .

14. A street vendor sells hamburgers, hot dogs, and 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 and

, what is the meaning of the dot product ? 15–20 Find the angle between the vectors. (First find an exact expression and then approximate to the nearest degree.)

15. ,

16. ,

17. ,

18. ,

19. ,

20. ,

aⴢ 共b ⫹ c兲

a

a

### ⱍ

ⴢ 共b ⫹ c兲

aⴢ b ⫹ c aⴢ b

b 具0.7, 1.2典 a 具⫺2, 3典

b 具⫺5, 12典 a具⫺2, 13

b 具2, 5, ⫺1典 a 具6, ⫺2, 3典

b 具6, ⫺3, ⫺8典 a具4, 1, 14

b 4i ⫹ 5k a 3i ⫹ 2j ⫺ k

2␲兾3 b

b

 5 a

a

 6

45⬚ b

b

 s6 a

a

### ⱍ

 3

uⴢ w uⴢ v

w

u

v

w

u v

iⴢ j  j ⴢ k  k ⴢ i  0 iⴢ i  j ⴢ j  k ⴢ k  1

c b

a

A 具a, b, c典 Aⴢ P P 具2, 1.5, 1典

b 具2, ⫺1典 a 具4, 3典

b 具5, 12典 a 具⫺2, 5典

b 具⫺2, 4, 3典 a 具3, ⫺1, 5典

b 具2, ⫺1, 0典 a 具4, 0, 2典

b 2i ⫺ k a 4i ⫺ 3j ⫹ k

b 4i ⫺ 3k a i ⫹ 2j ⫺ 2k

a 具s, 2s, 3s典 b 具t, ⫺t, 5t典 a i ⫺ 2 j ⫹ 3k b 5i ⫹ 9k

21–22 Find, correct to the nearest degree, the three angles of the triangle with the given vertices.

21. , ,

22. , ,

23–24 Determine whether the given vectors are orthogonal, parallel, or neither.

23. (a) ,

(b) ,

(c) ,

(d) ,

24. (a) ,

(b) ,

(c) ,

25. Use vectors to decide whether the triangle with vertices

, , and is right-angled.

26. Find the values of such that the angle between the vectors

, and is .

27. Find a unit vector that is orthogonal to both and . 28. Find two unit vectors that make an angle of with

.

29–30 Find the acute angle between the lines.

29. ,

30. ,

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. ,

32. , ,

33–37 Find the direction cosines and direction angles of the vector.

(Give the direction angles correct to the nearest degree.)

33. 34.

35. 36.

37. , where

38. If a vector has direction angles and , find the third direction angle .

P共2, 0兲 Q共0, 3兲 R共3, 4兲

A共1, 0, ⫺1兲 B共3, ⫺2, 0兲 C共1, 3, 3兲

a 具⫺5, 3, 7典 b 具6, ⫺8, 2典 a 具4, 6典 b 具⫺3, 2典

a ⫺i ⫹ 2 j ⫹ 5k b  3i ⫹ 4 j ⫺ k a 2i ⫹ 6 j ⫺ 4k b  ⫺3i ⫺ 9 j ⫹ 6k u 具⫺3, 9, 6典 v 具4, ⫺12, ⫺8典 u i ⫺ j ⫹ 2k v  2i ⫺ j ⫹ k u 具a, b, c典 v 具⫺b, a, 0典

P共1, ⫺3, ⫺2兲 Q共2, 0, ⫺4兲 R共6, ⫺2, ⫺5兲 x

i⫹ j i⫹ k 60⬚

v 具3, 4典

2x⫺ y  3 3x ⫹ y  7 x⫹ 2y  7 5x ⫺ y  2

y x2 y x3

y sin x y  cos x 0 艋 x 艋␲兾2

i⫺ 2j ⫺ 3k 12i⫹ j ⫹ k

␣  ␲兾4 ␤  ␲兾3

### 12.3Exercises

1. Homework Hints available at stewartcalculus.com

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SECTION 12.3 THE DOT PRODUCT 831 39–44 Find the scalar and vector projections of onto .

39. ,

40. ,

41. ,

42. ,

43. ,

44. ,

45. Show that the vector is orthogonal to . (It is called an orthogonal projection of .)

46. For the vectors in Exercise 40, find and illustrate by drawing the vectors , , , and .

47. If , find a vector such that .

48. Suppose that and are nonzero vectors.

(a) Under what circumstances is ? (b) Under what circumstances is ?

49. Find the work done by a force that moves an object from the point to the point 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 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?

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 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 N W in moving an object 5 m due west.

53. Use a scalar projection to show that the distance from a point

to the line is

Use this formula to find the distance from the point to

the line .

54. If , and , show

that the vector equation represents a sphere, and find its center and radius.

b a

a 具⫺5, 12典 b 具4, 6典 a 具1, 4典 b 具2, 3典

b 具1, 2, 3典 a 具3, 6, ⫺2典

b 具5, ⫺1, 4典 a 具⫺2, 3, ⫺6典

b j ⫹12k a 2i ⫺ j ⫹ 4k

b i ⫺ j ⫹ k a i ⫹ j ⫹ k

a orthab b ⫺ projab

b orthab

orthab projab

b a

compab 2 b

a 具3, 0, ⫺1典 b a

compab compba projab projba F 8 i ⫺ 6 j ⫹ 9k

30⬚

ax⫹ by ⫹ c  0 P1共x1, y1

a x1⫹ by1⫹ c

### ⱍ

sa2⫹ b2

b 具b1, b2, b3r 具x, y, z典, a  具a1, a2, a3

50⬚

55. Find the angle between a diagonal of a cube and one of its edges.

56. Find the angle between a diagonal of a cube and a diagonal of one of its faces.

57. A molecule of methane, , is structured with the four hydro- gen atoms at the vertices of a regular tetrahedron and the car- bon 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 . Hint: Take the vertices of the tetrahedron to be the points , ,

, and , as shown in the figure. Then the centroid is .

58. If , where , , and are all nonzero vectors, show that bisects the angle between and .

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:

62. The Triangle Inequality for vectors is

(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

and use Property 3 of the dot product.]

63. The Parallelogram Law states that

(a) Give a geometric interpretation of the Parallelogram Law.

(b) Prove the Parallelogram Law. (See the hint in Exercise 62.) 64. Show that if and are orthogonal, then the vectors

and must have the same length.

CH4

109.5⬚ [

(12, 12, 12) ]

H

H H

H C

x

y z

c

a

b

b

a a b c

c a b

aⴢ b

a

b

a⫹ b

a

b

a⫹ b

### ⱍ

2 共a ⫹ b兲 ⭈ 共a ⫹ b兲

a⫹ b

2

a⫺ b

2 2

a

2⫹ 2

b

### ⱍ

2

u⫹ v u⫺ v

u v

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838 CHAPTER 12 VECTORS AND THE GEOMETRY OF SPACE

SOLUTION

F r

F

r F

EXAMPLE 6

r

r

F

n

### ⬇ 9.66 n

FIGURE 5 n

75° 0.25 m 40 N

1–7 Find the cross product and verify that it is orthogonal to both a and b.

1. ,

2. ,

3. ,

4. ,

5. ,

6. ,

7. ,

8. If a苷 i ⫺ 2k and b 苷 j ⫹ k, find a ⫻ b. Sketch a, b, and a⫻ b as vectors starting at the origin.

9–12 Find the vector, not with determinants, but by using proper- ties of cross products.

9. 10.

11. 12.

13. State whether each expression is meaningful. If not, explain why. If so, state whether it is a vector or a scalar.

(a) (b)

(c) (d)

(e) (f )

a⫻ b

a苷 具6, 0, ⫺2典 b苷 具0, 8, 0典 a苷 具1, 1, ⫺1典 b苷 具2, 4, 6典 a苷 i ⫹ 3j ⫺ 2k b 苷 ⫺i ⫹ 5k a苷 j ⫹ 7k b 苷 2i ⫺ j ⫹ 4k a苷 i ⫺ j ⫺ k b 苷12i⫹ j ⫹12k

a苷 ti ⫹ cos tj ⫹ sin tk b 苷 i ⫺ sin tj ⫹ cos tk a苷 具t, 1, 1兾t典 b 苷 具t2, t2, 1典

aⴢ 共b ⫻ c兲 a⫻ 共b ⴢ c兲

a⫻ 共b ⫻ c兲 aⴢ 共b ⴢ c兲

14–15 Find and determine whether u⫻ v is directed into the page or out of the page.

14. 15.

16. The figure shows a vector in the -plane and a vector in the direction of . Their lengths are and

(a) Find .

(b) Use the right-hand rule to decide whether the com ponents of are positive, negative, or 0.

17. If and , find and .

18. If , , and , show that

.

19. Find two unit vectors orthogonal to both and .

u⫻ v

45°

| u |=4

| v |=5 | v |=16

120°

| u |=12

a xy b

k

a

b

a⫻ b

### ⱍ

a⫻ b

x

z

y b

a

a苷 具2, ⫺1, 3典 b苷 具4, 2, 1典 a⫻ b b⫻ a a苷 具1, 0, 1典 b 苷 具2, 1, ⫺1典 c苷 具0, 1, 3典 a⫻ 共b ⫻ c兲 苷 共a ⫻ b兲 ⫻ c

### 12.4Exercises

1. Homework Hints available at stewartcalculus.com

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SECTION 12.4 THE CROSS PRODUCT 839 20. Find two unit vectors orthogonal to both and .

21. Show that for any vector in . 22. Show that for all vectors and in . 23. Prove Property 1 of Theorem 11.

24. Prove Property 2 of Theorem 11.

25. Prove Property 3 of Theorem 11.

26. Prove Property 4 of Theorem 11.

27. Find the area of the parallelogram with vertices ,

, , and .

28. Find the area of the parallelogram with vertices ,

, , and .

29–32 (a) Find a nonzero vector orthogonal to the plane through the points , , and , and (b) find the area of triangle .

29. , ,

30. , ,

31. , ,

32. , ,

33–34 Find the volume of the parallelepiped determined by the vectors , , and .

33. , ,

34. , ,

35–36 Find the volume of the parallelepiped with adjacent edges , , and .

35. , , ,

36. , , ,

37. Use the scalar triple product to verify that the vectors

, , and

are coplanar.

38. Use the scalar triple product to determine whether the points

, , , and 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 .

j⫺ k i⫹ j V3

a 0⫻ a 苷 0 苷 a ⫻ 0

A共⫺2, 1兲 D共2, ⫺1兲

C共4, 2兲 B共0, 4兲

K共1, 2, 3兲 N共3, 7, 3兲

M共3, 8, 6兲 L共1, 3, 6兲

PQR R

Q P

R共5, 3, 1兲 Q共4, 1, ⫺2兲

P共0, ⫺2, 0兲

R共4, 3, ⫺1兲 Q共0, 5, 2兲

P共⫺1, 3, 1兲

P共1, 0, 1兲 Q共⫺2, 1, 3兲 R共4, 2, 5兲 P共0, 0, ⫺3兲 Q共4, 2, 0兲 R共3, 3, 1兲

c b a

PS PR PQ

S共3, 6, 1兲 R共1, 4, ⫺1兲

Q共2, 3, 2兲 P共⫺2, 1, 0兲

S共0, 4, 2兲 R共5, 1, ⫺1兲

Q共⫺1, 2, 5兲 P共3, 0, 1兲

w苷 5i ⫹ 9 j ⫺ 4 k v苷 3i ⫺ j

u苷 i ⫹ 5 j ⫺ 2 k

D共3, 6, ⫺4兲 C共5, 2, 0兲

B共3, ⫺1, 6兲 A共1, 3, 2兲

P

10° 70° 60 N

P a苷 具6, 3, ⫺1典 b苷 具0, 1, 2典 c苷 具4, ⫺2, 5典 a苷 i ⫹ j ⫺ k b 苷 i ⫺ j ⫹ k c 苷 ⫺i ⫹ j ⫹ k

40. Find the magnitude of the torque about if a 240-N force is applied as shown.

41. A wrench 30 cm long lies along the positive -axis and grips a bolt at the origin. A force is applied in the direction

at the end of the wrench. Find the magnitude of the force needed to supply of torque to the bolt.

42. Let v苷 5j and let u be a vector with length 3 that starts at the origin and rotates in the -plane. Find the maximum and minimum values of the length of the vector u⫻ v. In what direction does u⫻ v point?

43. If and , find the angle between

and .

44. (a) Find all vectors such that

(b) Explain why there is no vector such that

45. (a) Let be a point not on the line that passes through the points and . Show that the distance from the point to the line is

where QRl

and QPl .

(b) Use the formula in part (a) to find the distance from the point to the line through and

.

46. (a) Let be a point not on the plane that passes through the points , , and . Show that the distance from to the plane is

where QRl , QSl

, and QPl .

(b) Use the formula in part (a) to find the distance from the point to the plane through the points ,

, and .

47. Show that .

48. If , show that

y

xy

aⴢ b 苷 s3 a⫻ b 苷 具1, 2, 2典 a

b

v

P L

Q R d P

L

d

a⫻ b

a

### ⱍ

ab

P共1, 1, 1兲 Q共0, 6, 8兲

R共⫺1, 4, 7兲 P

Q R S d P

d

aⴢ 共b ⫻ c兲

a⫻ b

### ⱍ

abc

P共2, 1, 4兲 Q共1, 0, 0兲

R共0, 2, 0兲 S共0, 0, 3兲

a⫻ b

2

a

2

b

### ⱍ

2⫺ 共a ⴢ b兲2

a⫻ b 苷 b ⫻ c 苷 c ⫻ a a⫹ b ⫹ c 苷 0

P

30° 240 N

2 m 2 m

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840 CHAPTER 12 VECTORS AND THE GEOMETRY OF SPACE

49. Prove that .

50. Prove Property 6 of Theorem 11, that is,

51. Use Exercise 50 to prove that

52. Prove that

53. Suppose that .

(a) If , does it follow that ? (b) If , does it follow that ?

(c) If and , does it follow

that ? a苷 0

b苷 c aⴢ b 苷 a ⴢ c

b苷 c a⫻ b 苷 a ⫻ c

aⴢ b 苷 a ⴢ c a⫻ b 苷 a ⫻ c b苷 c

a⫻ 共b ⫻ c兲 苷 共a ⴢ c兲b ⫺ 共a ⴢ b兲c

a⫻ 共b ⫻ c兲 ⫹ b ⫻ 共c ⫻ a兲 ⫹ c ⫻ 共a ⫻ b兲 苷 0

## 冟

aaⴢ cⴢ d bbⴢ cⴢ d

## 冟

54. If , , and are noncoplanar vectors, let

(These vectors occur in the study of crystallography. Vectors of the form , where each is an integer, form a lattice for a crystal. Vectors written similarly in terms of

, , and form the reciprocal lattice.) (a) Show that is perpendicular to if .

(b) Show that for .

(c) Show that .

k1 k2 k3

ki vj i苷 j

kiⴢ vi苷 1 i苷 1, 2, 3 k1ⴢ 共k2⫻ k3兲 苷 1

v1ⴢ 共v2⫻ v3v3

v2

v1

k2v3⫻ v1 v1ⴢ 共v2⫻ v3k1v2⫻ v3

v1ⴢ 共v2⫻ v3

k3v1⫻ v2 v1ⴢ 共v2⫻ v3

ni

n1v1⫹ n2v2⫹ n3v3

### D I S C O V E RY P R O J E C TTHE GEOMETRY OF A TETRAHEDRON

A tetrahedron is a solid with four vertices, , , , and , and four triangular faces, as shown in the figure.

1. Let , , , and be vectors with lengths equal to the areas of the faces opposite the vertices , , , and , respectively, and directions perpendicular to the respective faces and pointing outward. Show that

2. The volume of a tetrahedron is one-third the distance from a vertex to the opposite face, times the area of that face.

(a) Find a formula for the volume of a tetrahedron in terms of the coordinates of its vertices , , , and .

(b) Find the volume of the tetrahedron whose vertices are , , , and .

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

(This is a three-dimensional version of the Pythagorean Theorem.)

P Q R S

v1 v2 v3 v4

P Q R S

v1⫹ v2⫹ v3⫹ v4苷 0 V

P Q R S

P共1, 1, 1兲 Q共1, 2, 3兲 R共1, 1, 2兲 S共3, ⫺1, 2兲

D2苷 A2⫹ B2⫹ C2 P

Q R

S

xy

L

P0

0

0

0

L L

P

L

v

P P0

r r0

L

### 12.5Equations of Lines and Planes

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848 CHAPTER 12 VECTORS AND THE GEOMETRY OF SPACE

1. Determine whether each statement is true or false.

(a) Two lines parallel to a third line are parallel.

(b) Two lines perpendicular to a third line are parallel.

(c) Two planes parallel to a third plane are parallel.

(d) Two planes perpendicular to a third plane are parallel.

(e) Two lines parallel to a plane are parallel.

(f ) Two lines perpendicular to a plane are parallel.

(g) Two planes parallel to a line are parallel.

(h) Two planes perpendicular to a line are parallel.

( i) Two planes either intersect or are parallel.

( j) Two lines either intersect or are parallel.

(k) 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 and parallel to the vector

3. The line through the point and parallel to the vector

4. The line through the point and parallel to the line , ,

5. The line through the point (1, 0, 6) and perpendicular to the plane

6–12 Find parametric equations and symmetric equations for the line.

6. The line through the origin and the point 7. The line through the points and 8. The line through the points and 9. The line through the points and

10. The line through and perpendicular to both and

11. The line through and parallel to the line

12. The line of intersection of the planes and

13. Is the line through and parallel to the

line through and ?

14. Is the line through and perpendicular to the

line through and ?

15. (a) Find symmetric equations for the line that passes through the point and is parallel to the vector

.

(b) Find the points in which the required line in part (a) inter- sects the coordinate planes.

1, 3, 23

x苷 ⫺1 ⫹ 2t

x⫹ 3y ⫹ z 苷 5

(0, 12, 1)

i⫹ j 共2, 1, 0兲

j⫹ k

x⫹ 2y ⫹ 3z 苷 1 x⫺ y ⫹ z 苷 1

16. (a) Find parametric equations for the line through that is perpendicular to the plane .

(b) In what points does this line intersect the coordinate planes?

17. Find a vector equation for the line segment from to .

18. Find parametric equations for the line segment from to .

19–22 Determine whether the lines and are parallel, skew, or intersecting. If they intersect, find the point of intersection.

19. : , ,

: , ,

20. : , ,

: , ,

21. :

:

22. :

:

23–40 Find an equation of the plane.

23. The plane through the point and perpendicular to the vector

24. The plane through the point and with normal vector

25. The plane through the point and with normal vector

26. The plane through the point and perpendicular to the

line , ,

27. The plane through the point and parallel to the plane

28. The plane through the point and parallel to the plane

29. The plane through the point and parallel to the plane

30. The plane that contains the line , , and is parallel to the plane

31. The plane through the points , , and 32. The plane through the origin and the points

and

L1 L2

L1 x苷 3 ⫹ 2t y 苷 4 ⫺ t z 苷 1 ⫹ 3t L2 x苷 1 ⫹ 4s y 苷 3 ⫺ 2s z 苷 4 ⫹ 5s L1 x苷 5 ⫺ 12t y 苷 3 ⫹ 9t

L2 x苷 3 ⫹ 8s y 苷 ⫺6s z 苷 7 ⫹ 2s L1

x⫺ 2

1 苷 y⫺ 3

⫺2 苷 z⫺ 1

⫺3 L2

x⫺ 3

1 苷 y⫹ 4

3 苷 z⫺ 2

⫺7 L1

x

1 苷 y⫺ 1

⫺1 苷 z⫺ 2 3 L2

x⫺ 2

2 苷 y⫺ 3

⫺2 苷 z 7

(⫺1, 12, 3)

i⫹ 4j ⫹ k

(1, 12, 13)

x⫹ y ⫹ z 苷 0

x苷 1 ⫹ t y 苷 2 ⫺ t

z苷 4 ⫺ 3t 5x⫹ 2y ⫹ z 苷 1

z苷 1 ⫺ 3t

### 12.5Exercises

1. Homework Hints available at stewartcalculus.com

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SECTION 12.5 EQUATIONS OF LINES AND PLANES 849 33. The plane through the points , , and

34. The plane that passes through the point and contains

the line , ,

35. The plane that passes through the point and contains

the line , ,

36. The plane that passes through the point and contains the line with symmetric equations

37. The plane that passes through the point and contains the line of intersection of the planes and

38. The plane that passes through the points and and is perpendicular to the plane

39. The plane that passes through the point and is perpen- dicular to the planes and

40. The plane that passes through the line of intersection of the planes and and is perpendicular to the plane

41–44 Use intercepts to help sketch the plane.

41. 42.

43. 44.

45–47 Find the point at which the line intersects the given plane.

45. , , ;

46. , , ;

47. ;

48. Where does the line through and intersect

the plane ?

49. Find direction numbers for the line of intersection of the planes and .

50. Find the cosine of the angle between the planes and .

51–56 Determine whether the planes are parallel, perpendicular, or neither. If neither, find the angle between them.

51. ,

52. ,

53. ,

54. ,

55. ,

56. ,

x苷 4 ⫺ 2t

y⫹ 2z 苷 3 x⫺ z 苷 1

x⫹ y ⫺ 2z 苷 1

3x⫹ y ⫹ 2z 苷 6 2x⫹ 5y ⫹ z 苷 10

6x⫹ 5y ⫺ 3z 苷 15 6x⫺ 3y ⫹ 4z 苷 6

x⫺ y ⫹ 2z 苷 9 z苷 5t

y苷 2 ⫹ t x苷 3 ⫺ t

x⫹ 2y ⫺ z ⫹ 1 苷 0 z苷 2 ⫺ 3t

y苷 4t x苷 1 ⫹ 2t

4x⫺ y ⫹ 3z 苷 8 x苷 y ⫺ 1 苷 2z

x⫹ y ⫹ z 苷 6

x⫹ z 苷 0 x⫹ y ⫹ z 苷 1

x⫹ y ⫹ z 苷 0 x⫹ 2y ⫹ 3z 苷 1

⫺3x ⫹ 6y ⫹ 7z 苷 0 x⫹ 4y ⫺ 3z 苷 1

3x⫺ 12y ⫹ 6z 苷 1 2z苷 4y ⫺ x

x⫺ y ⫹ z 苷 1 x⫹ y ⫹ z 苷 1

x⫹ 6y ⫹ 4z 苷 3 2 x⫺ 3y ⫹ 4z 苷 5

8y苷 1 ⫹ 2x ⫹ 4z x苷 4y ⫺ 2z

2 x⫺ y ⫹ 2z 苷 1 x⫹ 2y ⫹ 2z 苷 1

57. ,

58. ,

59–60 Find symmetric equations for the line of intersection of the planes.

59. ,

60. ,

61. Find an equation for the plane consisting of all points that are equidistant from the points and .

62. Find an equation for the plane consisting of all points that are equidistant from the points and .

63. Find an equation of the plane with -intercept , -intercept , and -intercept .

64. (a) Find the point at which the given lines intersect:

(b) Find an equation of the plane that contains these lines.

65. Find parametric equations for the line through the point that is parallel to the plane and perpendicular to the line , , . 66. Find parametric equations for the line through the point

that is perpendicular to the line , , and intersects this line.

67. Which of the following four planes are parallel? Are any of them identical?

68. Which of the following four lines are parallel? Are any of them identical?

, ,

, ,

69–70 Use the formula in Exercise 45 in Section 12.4 to find the distance from the point to the given line.

69. ; , ,

70. ; , ,

3x⫺ 2y ⫹ z 苷 1 2x ⫹ y ⫺ 3z 苷 3

5x⫺ 2y ⫺ 2z 苷 1 4x ⫹ y ⫹ z 苷 6 z苷 2x ⫺ y ⫺ 5 z 苷 4x ⫹ 3y ⫺ 5

x a y b

z c

r苷 具1, 1, 0典 ⫹ t 具1, ⫺1, 2典 r苷 具2, 0, 2典 ⫹ s具⫺1, 1, 0典

x苷 1 ⫹ t y 苷 1 ⫺ t z 苷 2t

y苷 1 ⫺ t z 苷 2t

P1: 3x⫹ 6y ⫺ 3z 苷 6 P2: 4x⫺ 12y ⫹ 8z 苷 5 P3: 9y苷 1 ⫹ 3x ⫹ 6z P4: z苷 x ⫹ 2y ⫺ 2

L1: x苷 1 ⫹ 6t y 苷 1 ⫺ 3t z 苷 12t ⫹ 5 L2: x苷 1 ⫹ 2t y 苷 t z 苷 1 ⫹ 4t L3: 2x⫺ 2 苷 4 ⫺ 4y 苷 z ⫹ 1 L4: r苷 具3, 1, 5典 ⫹ t 具4, 2, 8典

x⫹ 2y ⫹ 2z 苷 1 x⫹ y ⫹ z 苷 1

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850 CHAPTER 12 VECTORS AND THE GEOMETRY OF SPACE 71–72 Find the distance from the point to the given plane.

71. ,

72. ,

73–74 Find the distance between the given parallel planes.

73. ,

74. ,

75. Show that the distance between the parallel planes and is

76. Find equations of the planes that are parallel to the plane and two units away from it.

77. Show that the lines with symmetric equations and are skew, and find the distance between these lines.

2x⫺ 3y ⫹ z 苷 4 4x ⫺ 6y ⫹ 2z 苷 3 6z苷 4y ⫺ 2x 9z 苷 1 ⫺ 3x ⫹ 6y

ax⫹ by ⫹ cz ⫹ d1苷 0 ax⫹ by ⫹ cz ⫹ d2苷 0

D

d1⫺ d2

### ⱍ

sa2⫹ b2⫹ c2

x⫹ 2y ⫺ 2z 苷 1

x苷 y 苷 z x⫹ 1 苷 y兾2 苷 z兾3

3x⫹ 2y ⫹ 6z 苷 5 共1, ⫺2, 4兲

x⫺ 2y ⫺ 4z 苷 8 共⫺6, 3, 5兲

78. Find the distance between the skew lines with parametric

equations , , , and ,

, .

79. Let be the line through the origin and the point . Let be the line through the points and . Find the distance between and .

80. Let be the line through the points and . Let be the line of intersection of the planes and ,

where is the plane and is the plane

through the points , , and . Calculate the distance between and .

81. If , , and are not all 0, show that the equation

represents a plane and is a normal vector to the plane.

Hint: Suppose and rewrite the equation in the form

82. Give a geometric description of each family of planes.

(a) (b)

(c)

L1 共2, 0, ⫺1兲

L2 共1, ⫺1, 1兲 共4, 1, 3兲

L1 L2

L1 共1, 2, 6兲 共2, 4, 8兲

L212

1 x⫺ y ⫹ 2z ⫹ 1 苷 02

a b c

ax⫹ by ⫹ cz ⫹ d 苷 0 具a, b, c典

a苷 0

a

x da

## 冊

⫹ b共y ⫺ 0兲 ⫹ c共z ⫺ 0兲 苷 0

x⫹ y ⫹ z 苷 c x⫹ y ⫹ cz 苷 1

y cos ␪ ⫹ z sin ␪ 苷 1

x苷 1 ⫹ 2s z苷 2t

y苷 1 ⫹ 6t x苷 1 ⫹ t

z苷 ⫺2 ⫹ 6s y苷 5 ⫹ 15s

### L A B O R AT O RY P R O J E C TPUTTING 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 rect- angular 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 -plane with vertices

and , and the camera is placed at . A line in the scene passes through the points and . At what points should 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 , , , and

is added to the scene. The line intersects this rectangle. To make the rect- angle appear opaque, a programmer can use hidden line rendering, which removes portions of objects that are behind other objects. Identify the portion of that should be removed.

yz 共0, ⫾400, 0兲

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Updating...

## References

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