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Complete cheat sheet on Analytic Geometry in Three Dimensions with examples, exercises and problems solving
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S E L E C T E D A P P L I C AT I O N S
11.1 The Three-Dimensional Coordinate System
11.2 Vectors in Space
11.3 The Cross Product of Two Vectors
11.4 Lines and Planes in Space
Analytic Geometry
in Three Dimensions
Arnold Fisher/Photo Researchers, Inc.
811
The Three-Dimensional Coordinate System
x
y
z
x
y
yz -plane
xy -plane
xz -plane
z
( x , y , z )
x , y , z ,
812 Chapter 11 Analytic Geometry in Three Dimensions
What you should learn
Why you should learn it The three-dimensional coordinate system can be used to graph equations that model surfaces in space, such as the spherical shape of Earth, as shown in Exercise 66 on page 819.
The Three-Dimensional Coordinate System
NASA
11.
x
z
y x
z
y x
z
y x
z
y x
z
y x
z
y x
z
y x
z
y
Using the Midpoint Formula in Space
Solution
The Equation of a Sphere
Equation of sphere in space
Equation of circle in the plane
x h ^2 y k ^2 r^2
x h ^2 y k ^2 z j ^2 r^2
x h ^2 y k ^2 z j ^2 r.
x , y , z x , y , z h , k , j r ,
h , k , j r
Midpoint: ( 52 , 1,^72 )
x
y
2 1
4 3
− 1
− (^4) − 3 − 2
− 3
4
(^3 2 ) 4
2
z
2 ^
(^)
2
5, 2, 3 0, 4, 4.
814 Chapter 11 Analytic Geometry in Three Dimensions
Example 3
Standard Equation of a Sphere
x h ^2 y k ^2 z j ^2 r^2.
h , k , j r
( , x y z , )
( , h k j , )
r
Sphere: radius r ; center ( , h k j , )
x
y
z
Finding the Equation of a Sphere
Solution
Standard equation
Substitute.
Finding the Center and Radius of a Sphere
Solution
1, 2, 3,^ 6.
1
3 2
2
1 3 4 5
1
5 4
3 6 7
r = 3
x
y
z
xy 2, 4, 0.
x 2 ^2 y 4 ^2 z 3 ^2 32
x h ^2 y k ^2 z j ^2 r^2
2, 4, 3
Section 11.1 The Three-Dimensional Coordinate System 815
Example 4
Example 5
x 1 ^2 y 2 ^2 z 3 ^2 6
2
x^2 2 x 1 y^2 4 y 4 z^2 6 z 9 8 1 4 9
x^2 2 x (^) y^2 4 y (^) z^2 6 z (^) 8
1, 4, 2
3, 2, 6
Exploration
r = 6
x
1 2
5 4 3 3
1
5
6
y
z
Section 11.1 The Three-Dimensional Coordinate System 817
In Exercises 1 and 2, approximate the coordinates of the points.
1. 2.
In Exercises 3–6, plot each point in the same three-dimen- sional coordinate system.
3. (a) 4. (a) (b) (b) 5. (a) 6. (a) (b) (b)
In Exercises 7–10, find the coordinates of the point.
7. The point is located three units behind the -plane, three units to the right of the -plane, and four units above the -plane. 8. The point is located six units in front of the -plane, one unit to the left of the -plane, and one unit below the -plane. 9. The point is located on the -axis, 10 units in front of the -plane. 10. The point is located in the -plane, two units to the right of the -plane, and eight units above the -plane.
In Exercises 11–16, determine the octant(s) in which is located so that the condition(s) is (are) satisfied.
**11.
16.**
In Exercises 17–24, find the distance between the points.
**17.
23.**
24. 2, 4, 0,0, 6, 3
yz > 0
xy < 0
y < 0
z > 0
x < 0, y > 0, z < 0
x > 0, y < 0, z > 0
x, y, z
xz xy
yz
yz
x
xy
xz
yz
xy
xz
yz
x y
543
2
2
4
3 − 5 − 6
− (^4) − 3 − 2
z
x^ y B
4
3 2
4 3 2
−^ −^344
− (^4) − (^3) − 2 − 4
z
11.1 Exercises
1. A _______ coordinate system can be formed by passing a -axis perpendicular to both the -axis and the -axis at the origin. 2. The three coordinate planes of a three-dimensional coordinate system are the _______ , the _______ , and the _______. 3. The coordinate planes of a three-dimensional coordinate system separate the coordinate system into eight _______. 4. The distance between the points and can be found using the _______ _______ in Space. 5. The midpoint of the line segment joining the points and given by the Midpoint Formula in Space is _______. 6. A _______ is the set of all points such that the distance between and a fixed point is 7. A _______ in _______ is the collection of points satisfying an equation involving and 8. The intersection of a surface with one of the three coordinate planes is called a _______ of the surface.
x , y , z.
h , k , j r.
x , y , z x , y , z
x 1 , y 1 , z 1 x 2 , y 2 , z 2
x 1 , y 1 , z 1 x 2 , y 2 , z 2
x y
z
In Exercises 25–28, find the lengths of the sides of the right triangle. Show that these lengths satisfy the Pythagorean Theorem.
25. 26.
In Exercises 29 and 30, find the lengths of the sides of the triangle with the indicated vertices, and determine whether the triangle is a right triangle, an isosceles triangle, or neither.
29. 30.
In Exercises 31–38, find the midpoint of the line segment joining the points.
**31. 32.
38.**
In Exercises 39–46, find the standard form of the equation of the sphere with the given characteristics.
39. 40. 41. Center: radius: 3 42. Center: radius: 6 43. Center: diameter: 10 44. Center: diameter: 8 45. Endpoints of a diameter: 46. Endpoints of a diameter:
In Exercises 47–56, find the center and radius of the sphere.
**47.
56.**
In Exercises 57– 60, sketch the graph of the equation and sketch the specified trace.
57. -trace 58. -trace 59. -trace 60. -trace
In Exercises 61 and 62, use a three-dimensional graphing utility to graph the sphere.
61. 62.
63. Crystals Crystals are classified according to their symmetry. Crystals shaped like cubes are classified as isometric. The vertices of an isometric crystal mapped onto a three-dimensional coordinate system are shown in the figure. Determine
64. Crystals Crystals shaped like rectangular prisms are classified as tetragonal. The vertices of a tetragonal crystal mapped onto a three-dimensional coordinate system are shown in the figure. Determine 65. Architecture A spherical building has a diameter of 165 feet. The center of the building is placed at the origin of a three-dimensional coordinate system. What is the equation of the sphere?
x , y , z .
x
y
z
( x , y , z )
x
y
z
( x , y , z )
x , y , z .
x^2 y^2 z^2 6 y 8 z 21 0
x^2 y^2 z^2 6 x 8 y 10 z 46 0
x^2 y 1 ^2 z 1 ^2 4; xy
x 2 ^2 y 3 ^2 z^2 9; yz
x^2 y 3 ^2 z^2 25; yz
x 1 ^2 y^2 z^2 36; xz
4 x^2 4 y^2 4 z^2 4 x 32 y 8 z 33 0
9 x^2 9 y^2 9 z^2 6 x 18 y 1 0
2 x^2 2 y^2 2 z^2 2 x 6 y 4 z 5 0
9 x^2 9 y^2 9 z^2 18 x 6 y 72 z 73 0
x^2 y^2 z^2 8 y 6 z 13 0
x^2 y^2 z^2 4 x 8 z 19 0
x^2 y^2 z^2 6 x 4 y 9 0
x^2 y^2 z^2 4 x 2 y 6 z 10 0
x^2 y^2 z^2 8 y 0
x^2 y^2 z^2 5 x 0
2
2
3
3 (^3 ) (^5 )
6 4 2
r = 2
y
x
z
65 4 5 6
r = 4
x^ y
z
2 3
2
4
− 3 ( 2, 5, 0)− (^) y
x
z
3
(^2 )
4
− 3 − 4
− (^3) − (^2)
− 4
(0, 4, 0) (^) y
x
z
818 Chapter 11 Analytic Geometry in Three Dimensions
What you should learn
Why you should learn it Vectors in space can be used to represent many physical forces, such as tension in the cables used to support auditorium lights, as shown in Exercise 44 on page 826.
Vectors in Space
Component form
Unit vector form
Q q ( (^) 1 , q (^) 2 , q 3 )
P p ( (^) 1 , p (^) 2 , p 3 ) v
x
y
z
〈 v^ 1 ,^ v^ 2 , v 3 〉
i
k j
x
y
z
P p 1 , p 2 , p 3 Q q 1 , q 2 , q 3 ,
820 Chapter 11 Analytic Geometry in Three Dimensions
Vectors in Space
SuperStock
11.
Vectors in Space
Vector addition
Scalar multiplication
u (^) v u 1 v 1 u 2 v 2 u 3 v 3. Dot product
Finding the Component Form of a Vector
Solution
Finding the Dot Product of Two Vectors
Solution
v − u
v
u
Origin
θ
0, 3, 2 (^) 4, 2, 3 0 4 3 2 2 3
0, 2, 2 (^) 0,
2 ^
(^) 0,
2
3, 4, 2 3, 6, 4. v.
Section 11.2 Vectors in Space 821
Some graphing utilities have the capability to perform vector operations, such as the dot product. Consult the user’s guide for your graphing utility for specific instructions.
Te c h n o l o g y
Example 1
Example 2
Angle Between Two Vectors
u v
Using Vectors to Determine Collinear Points
Solution
Finding the Terminal Point of a Vector
Solution
Q 7, 1, 5.
**
Q q 1 , q 2 , q 3 ,
P 3, 1, 6
v 4, 2, 1 P 3, 1, 6.
− 2 2
2
4
4
− 10 − 8 − 6 − 4
x
y
z
**
**
** 4 2, 11 1 , 0 4 6, 10, 4 .
** 5 2, 4 1 , 6 4 3, 5, 2
**
**
P 2, 1, 4, Q 5, 4, 6, R 4, 11, 0
**
Section 11.2 Vectors in Space 823
Example 5
Example 6
Application
Solving an Equilibrium Problem
Solution
**
**
2 0, 0 2, 0 1
z
3
**
**
0 0, 4 2, 0 1
v 0,
5
**
**
2 0, 0 2, 0 1
u (^)
3
Q 0, 4, 0, R 2, 0, 0.
S 0, 2, 1 . P 2, 0, 0,
824 Chapter 11 Analytic Geometry in Three Dimensions
u
z
w
v
1
4
3
1 4 2
3
4
(^21) 3
3 4
1
2
3
x
y
z
Example 7
In Exercises 27–30, determine whether u and v are orthog- onal, parallel, or neither.
27. 28.
In Exercises 31–34, use vectors to determine whether the points are collinear.
31.
32.
33.
34.
In Exercises 35–38, the vector v and its initial point are given. Find the terminal point.
35. 36.
Initial point: Initial point:
37. 38.
Initial point: Initial point:
39. Determine the values of such that where 40. Determine the values of such that where
In Exercises 41 and 42, write the component form of v.
41. v lies in the -plane, has magnitude 4, and makes an angle of with the positive -axis. 42. v lies in the -plane, has magnitude 10, and makes an angle of 60 with the positive -axis. 43. Tension The weight of a crate is 500 newtons. Find the tension in each of the supporting cables shown in the figure.
Synthesis
True or False? In Exercises 45 and 46, determine whether the statement is true or false. Justify your answer.
45. If the dot product of two nonzero vectors is zero, then the angle between the vectors is a right angle. 46. If and are parallel vectors, then points and are collinear. 47. What is known about the nonzero vectors and if Explain. 48. Writing Consider the two nonzero vectors and Describe the geometric figure generated by the terminal points of the vectors and where and represent real numbers.
Skills Review
In Exercises 49–52, find a set of parametric equations for the rectangular equation using (a) and (b)
51. y x^2 8 52. y 4 x^3
y
x
y 3 x 2
t x t x 1.
t
t v , u t v , s u t v , s
u v.
u (^) v < 0?
u v
x y A
C D 60 cm
45 cm 70 cm
65 cm
115 cm
z
z
xz
45 y
yz
u 2 i 2 j 4 k.
c c u 12,
u i 2 j 3 k.
c c u 3,
2, 1,^ ^32 3, 2,^ ^12
v (^) 4, 32 , ^14 v (^) 52 , ^12 , 4
v 2, 4, 7 v 4, 1, 1
v 4 i 10 j k v 8 i 4 j 8 k
u 34 i 12 j 2 k u i 12 j k
v 8, 4, 10 v 2, 1, 5
u 12, 6, 15 u 1, 3, 1
826 Chapter 11 Analytic Geometry in Three Dimensions
disks of radius 18 inches. Each disk is supported by three equally spaced cables that are inches long (see figure).
(a) Write the tension in each cable as a function of Determine the domain of the function. (b) Use the function from part (a) to complete the table.
(c) Use a graphing utility to graph the function in part (a). What are the asymptotes of the graph? Interpret their meaning in the context of the problem. (d) Determine the minimum length of each cable if a cable can carry a maximum load of 10 pounds.
18 in.
Model It
Section 11.3 The Cross Product of Two Vectors 827
The Cross Product
b 2
u 2 v 3 u 3 v 2 i u 1 v 3 u 3 v 1 j u 1 v 2 u 2 v 1 k
v 3
v 3
v 2
v 3
What you should learn
Why you should learn it
The cross product of two vectors in space has many applications in physics and engineering. For instance, in Exercise 43 on page 833, the cross product is used to find the torque on the crank of a bicycle’s brake.
The Cross Product of Two Vectors
Carl Schneider/Getty Images
11.
Definition of Cross Product of Two Vectors in Space
u v u 2 v 3 u 3 v 2 i u 1 v 3 u 3 v 1 j u 1 v 2 u 2 v 1 k.
Exploration
Geometric Properties of the Cross Product
Using the Cross Product
Solution
u v 6 ^2 3 ^2 62
(^0)
Section 11.3 The Cross Product of Two Vectors 829
2
− 2
− 4
− 6
2 x 4
y
2
4
6
8
4 6
u
v
u ⋅ v
z
j
i
k = i × j
xy -plane
x
y
z
Geometric Properties of the Cross Product
Example 2
Geometric Application of the Cross Product
Solution
**
**
**
**
**
**
**
**
**
**
(^6)
**
**
**
**
**
**
**
**
**
**
**
**
**
**
A 5, 2, 0, B 2, 6, 1, C 2, 4, 7, D 5, 0, 6
830 Chapter 11 Analytic Geometry in Three Dimensions
Example 3
If you connect the terminal points of two vectors u and v that have the same initial points, a triangle is formed. Is it possible to use the cross product u v to determine the area of the triangle? Explain. Verify your conclusion using two vectors from Example 3.
Exploration
x
y
4
8
(^6 )
6
8
z