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Vectors and Motion In Space

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Title: Vectors and Motion In Space


1
Vectors and Motion In Space
Chapter 10
  • 170 121 Engineering Mathematics II
  • 16 ????????? 2547

2
Content
?????????????? Cartesian Coordinate, vector ???
operations ?????????, ????????????????????????, ??
????????????, vector-valued functions ??? curve
???????, ?????????? curve ??? unit vector
???????????? curve ????? Space
(???????????? ?????) ?????????????????????????????
??? ????? ????? ?????????? ???????????????????????
???? ????????????????????? Three Dimensional
Space ?????????????????????????????????????????
????????????????? (???????? Space
?????????????????????????? ???????????????????????
????????? ???????????????? ???????????????????????
?????????????????????????????????????????? ???????
?????? Space ????????????????)
3
Cartesian Coordinate
??????????????? 3 ???? ?????????? Cartesian
Coordinate ?????????? ??????? 3 ??? (x,y,z)
??????????????????????? X, Y ??? Z
???????? ?????????????? Origin (0,0,0) ??? X
??? Y ?????? Z ????????????????????? (Right Hand
System) ?????? ???????? ????????????????? 4
?????????? X ???????? Y ???? ?????????????????? ??
?????? Z
4
Cartesian Coordinate
Cartesian Coordinate ?????????????????????
Rectangular Coordinate ???????????? ??? X, Y, Z
???????????????????? (x,y,z) ??????????????????? ?
????????? (Rectangular box) ??????????????????????
? (0,0,0)
Z
P(x,y,z)
Y
(0,0,0)
X
5
Octants
?????????????? XYZ ?????????????? (Reference
frame) ????????????? Space ??????????? x0, ?????
y0 ???????? z0 ??????? 8 ???????????? ????????
Octant ??????? x ???? , y ???? , z ????
???????? First Octant
Z
????? x0 (???????? YZ plane)
????? y0 (???????? XZ plane)
Y
????? z0 (???????? XY plane)
X
6
Cylinder
?? 3 ????????????????????????????????????? ????
???? Z ??????????????????????
??? R ????????????????????
??????????????????????????? ????????????????? ???
z ?????? ????????????????? ?????? z
??????????????? ????????????????? z
??????????? ?????????? ????? 1 ????
??????????????? ??????? z ?????????????????? ?????
?????????
(????????????? Thomas Calculus)
7
Component Form of a Vector in Space
??????????????? vector ?? Space 3
????????????????? ???? ??? Component Form
Z
v1, v2, v3 ???????????????? ????????
X,Y, Z ????????
Y
X
8
Standard Unit Vectors For 3D Space
??????????????? vector ?????? standard unit
vector ??????
??? ??? standard Unit
vectors ????????? X, Y, Z ???????? ??????
Z
Y
X
9
Position Vector
??????????? vector ??????????????? ????????
vector ?????? Position vector
????
???? Position vector ?????? P(x,y,z)
(????????????? Thomas Calculus)
10
Directed Line Segment
(????????????? Thomas Calculus)
11
Magnitude of Vectors
??????? vector ?????????????????????
(????????????? Thomas Calculus)
12
Unit Vector
???????????????????? vector ?????????? unit
vector ?????? 1 ????
?????????????? vector ????????
????????????? ?????????
???????
?????????
13
Dot Product
?????
???? operation ??????? vector ??? vector
???????????? Scalar ????????????????????? vector
?????????
??? ?????????? ?????
14
Properties of Dot Product
1.
2.
3.
4.
5.
15
Vector Projection
Projection of onto
???????????????????? ????????????????
(????????????? Thomas Calculus)
16
Vector Project
????????????? ??????? 2 ??????????
????????????? ??????????
????????????? ?????????????
(????????????? Thomas Calculus)
17
Cross Product
?????
???? Operation ??????? ???
???????????? vector ???????????????????????
???
(?????????????????)
(????????????? Thomas Calculus)
18
Cross Product
???????
????????????????????????????????
(Parallelogram) ??????????????????? ???
(????????????? Thomas Calculus)
19
Properties of Cross Product
1.
(????????????? Thomas Calculus)
20
Properties of Cross Product
2.
3.
4.
5.
21
Cross Products of Standard Unit Vectors
???
-
??
-
-
22
Determinant Formula for Cross Product
23
Application of Cross Product Torque
Torque ???????????????????????????????????????????
???? ???????????
(????????????? Thomas Calculus)
24
Box Product (Triple Scalar Product)
????? Box product ???????
(?????????????) ????????
?????????????????????? parallelpiped ??????
(????????????? Thomas Calculus)
25
Formula for Box Product
Box product ???????????????????????? Determinant
??????
???????????? Box product
26
Lines in Space
???????????????????????? ???????????????????
P0 ????????????????? ????????? v ??????????
(????????????? Thomas Calculus)
???????????????????????????? P0 ??? P ???????????
vector ??????? v
????????
t ??????? scalar ????????
27
Equations for a Line in Space
???
???
??????
???????????? Parametric Equations for a Line
??????
??????????????????????????????????????????????????
????
????????????? vector equation ????????
???
Position vector
28
Example Lines in Space
????????????????????????????? (-2,0,4) ??? (0,4,2)
?????? 1. ??????????? 2. ??????????
0
(????????????? Thomas Calculus)
29
Distance Between a Point and a Line
????????????? S ?????????? L ???????????
L
30
Planes in Space
?????????????????????????????????????????????
??? P0 ????????????????? Vector
?????????????????? (Normal vector)
(????????????? Thomas Calculus)
????????? ??????????????? ??????? Dot
product ??????? vector ????? ????????
?????? 0 ????
31
Equations for a Plane in Space
??? P0(x0,y0,z0) ??? P(x,y,z) ????????????????? M
????????
???? vector ??????? M ????????????????????????????
????
?????
???? normal vector ???????? M
32
Equations for a Plane in Space
?????????????
??????????????????
?????
????
???
33
Example a Plane in Space
?????????????????????????????? A(0,0,1), B(2,0,0)
??? C(0,3,0)
?????? 1. ????? vector 2 ????????????????????
2. ????? normal vector ???????? 3. ??????

34
Example a Plane in Space
??????????????? A(0,0,1), B(2,0,0) ??? C(0,3,0)
(????????????? Thomas Calculus)
35
Lines of Intersection Between Planes
?????????????????????? v ????????????????????????
???????????????????
(????????????? Thomas Calculus)
36
Example a Line of Intersection Between Planes
??????????????????????????????????????????????????
1. ????????????????????? 2. ????????????????????
???? 3. ?????????????????
37
Distance Between a point and a Plane
????????????? S ???????? M ???????????
q
M
(????????????? Thomas Calculus)
38
Example Distance Between a point and a Plane
39
Cylinders
?????????????????? Cylinder ??????????????
???????????????? ??????????? Cylinder
????????????????????????????????????????? ???
Cylinder ??????? ????????????????????????????????
?????????????????????????????????????
????????
????? ?? space 3
?????????????
???????
??????? ??? z ???? ?????? x0 ???????????
(????????????? Thomas Calculus)
40
Example Cylinder
???????????????
(????????????? Thomas Calculus)
41
Cylinders
???????????????????? - ??????????? Cylinder
????????? f(x,y) c ???????? Cylinder
????????? ???????????????????????? Z -
??????????? Cylinder ????????? f(x,z) c
???????? Cylinder ????????? ??????????????????????
?? Y ?????????? ???????? 2 ????????????
Cartesian coordinate 3 ???? ??????????? Cylinder
??????????????????????????????
???????????? Cylinder ????????????????????????????
???????????????? Cylinder ???????????????????
42
Example Cylinder
?????????? Cylinder
(?????????? X)
(????????????? Thomas Calculus)
43
Quadric Surfaces
Quadric Surface ???? graph ?? Space
????????????? 2 ????????? x,y, ???z
???????????????????????
(1)
??? A, B, C, D, E, F, G, H, J ??? K ???????????
???????? 2 ????????????????????? Ellipses,
Parabolas ??? Hyperbolas ???????????? ?? Space 3
????????? Quadric Surfaces ??????????? Ellipsoids,
Paraboloids, Elliptic Cones ???
Hyperboloids ???????????? (Sphere) ???????
Ellipsoid ????????
44
Analysis of Quadric Surface
??????????????????????????????????????? Quadric
surface ?????????????? ???????? ?????????????
Graph ?????????????????????? Quadric surface
???????? XY, ????? YZ ???????? XZ
(????????????????????? Coordinate
planes) ?????????????? Graph ????????????????????
?? Surface ???????? YZ ????????? ?????????????????
???? x ?????????? Quadric surface ????
0 ?????????????? Graph ??????????????????????
Surface ???????? XY ????????? ????????????????????
? z ?????????? Quadric surface ????
0 ?????????????? Graph ??????????????????????
Surface ???????? XZ ????????? ????????????????????
? y ?????????? Quadric surface ???? 0 Graph
????????????????????????????? Quadric Surface
?????? Curve ??????????????????
45
Ellipsoid
?????????????? Ellipsoid ????????????????????
(0,0,0) ???
(2)
c
Ellipsoid ???????????????????? (0,0,0)
???? ????????? X ??? (a,0,0) ???
(-a,0,0) ??????????? Y??? (0,b,0) ???
(0,-b,0) ??????????? Z ??? (0,0,c) ??? (0,0,-c)
b
a
-c
(????????????? Thomas Calculus)
46
Cross Sections of an Ellipsoid
????? set ??? z z0 ??????
????? set ??? z 0 ??????????
????? set ??? x 0 ??????????
(????????????? Thomas Calculus)
47
Cross Sections of an Ellipsoid
????????? Cross sections ??? Ellipsoid ???
Coordinate planes ?????????????????? (Ellipse)
????
?????? Cross section ??? Ellipsoid ???????? z
z0 ????????
??????????????
?????????????????????????
?????? a,b,c ???????? Ellipsoid
??????????????????????????????? Ellipsoid of
Revolution ????????? a,b,c ???????????????????????
????? (Sphere)
48
Elliptic Paraboloid
?????????????? Elliptic Paraboloid
??????????????? X ??? Y ???
(3)
??????????????? Origin (0,0,0) ???????????
Vertex ??? Graph ???????? z ?????????? Paraboloid
???????????????????????????????? ????????????
????????????????????????? Elliptic Paraboloid
???????? a b
(0,0,0)
(????????????? Thomas Calculus)
49
Cross Sections of Paraboloid
????? set ??? y 0 ??????
????? set ??? z c ??????????
????? set ??? x 0 ??????
(????????????? Thomas Calculus)
50
Cross Sections of Paraboloid
- Cross section ??????????????? z z0, z0 gt 0
??? Paraboloid ?????????? 3 ????????? Ellipse
???? - ??????????????????????? z ??????
??????????? Paraboloid ?????????? 3 ?????
Parabola ????
51
Cone
?????????????? Elliptic Paraboloid ???????? z
?????????????? ???
(4)
??????????????? Origin (0,0,0) ?????????????
?? X ??? Y ?????? a ??? b ???????
???????????? ??????????? Circular cone
(????????????? Thomas Calculus)
52
Cross Sections of a Cone
????? set ??? z c ??????????
1
????? set ??? x 0 ??????????????? 2 ????
????? set ??? y 0 ??????????????? 2 ????
????? set ??? z 0 ????????????????? z
?????????????? Origin (0,0,0)
(????????????? Thomas Calculus)
53
Cross Sections of a Cone
?????????????????? ????? set ??? z z0 ??? z0
?????????? 0, ??????????
????? set ??? x x0 ??? x0 ?????????? 0 ??????
????? set ??? y y0 ??? y0 ?????????? 0 ??????
54
Cross Sections of a Cone
- Cross section ??????????????? z z0, z0
?????????? 0, ?????????????? ??? 4 ?????????
Ellipse ???? - ??????????????????????? z
?????? ????????????????????????? 4 ???????????? 2
??????????????????? (0,0,0) ????
55
Hyperboloid
Hyperboloid ?? 2 ????
Hyperboloid of One Sheet
Hyperboloid of Two Sheets
(????????????? Thomas Calculus)
56
Hyperboloid of One Sheet
?????????????? Hyperboloid of One Sheet
??????????????? Z ???
(5)
???????????????????????????????
???????????? Coordinate planes ??????? ??? a b
???????? Hyperboloid ?????? Surface of Revolution
(????????????? Thomas Calculus)
57
Cross Sections of Hyperboloid of One Sheet
????? y 0 ??????
????? z c ??????
????? z 0 ??????
????? x 0 ??????
(????????????? Thomas Calculus)
58
Cross Sections of Hyperboloid of One Sheet
- Cross section ??????????????? z z0 ???
Hyperboloid ??????? ??? 5 ????????? Ellipse ????
??? a b ?????????????????????????? -
??????????????????????? z ?????? ???????????
Hyperboloid ?????????? 5 ????? Hyperbola ????
59
Hyperboloid of Two Sheets
?????????????? Hyperboloid of Two Sheets
??????????????? Z ???
(6)
?????????????????????? 2 ?????????? ????????
??????? coordinate planes ??????? ????????????????
??? XY ??? (0,0,c) ??? (0,0,-c) ???????
Vertices ??? Graph
(0,0,c)
(0,0,-c)
(????????????? Thomas Calculus)
60
Cross Sections of Hyperboloid of Two Sheets
????? y 0 ??????
????? x 0 ??????
(????????????? Thomas Calculus)
61
Cross Sections of Hyperboloid of Two Sheets
- Cross section ??????????????? z z0, z0 gt c,
??? Hyperboloid ?????????? 6 ????????? Ellipse
???? ??? a b ?????????????????????????? -
??????????????????????? z ?????? ???????????
Hyperboloid ?????????? 6 ????? Hyperbola ????
62
Hyperboloids
Both hyperboloids are asymptotic to cone
??????????? 5 ??? 6 ????????????? ??? 1
????????????????????? 0 ????????????????????
??????????? Hyperboloid ???????????
(???????) ????????????
(????????????? Thomas Calculus)
63
Hyperbolic Paraboloid a Saddle
???????????????? (Saddle) ???? Hyperbolic
Paraboloid ?????????????
(7)
??? c gt 0
Saddle ???????????? ????? x0 (?????
YZ) ???????? y 0 (????? XZ)
(????????????? Thomas Calculus)
64
Cross Sections of a Saddle
????? x 0 ??????
????? z c ??????
????? z -c ??????
(????? y 0)
(????????????? Thomas Calculus)
65
Cross Sections of a Saddle
- Cross section ??????????????? z 0 ??????
Saddle ?????????? 7 ????????????? 2
??????????????????? (0,0,0) - Cross section
??????????????? z z0, z0 ?????????? 0, ??????
Saddle ?????????? 7 ????????? Hyperbola -
?????????????????? X ??????????? Saddle
?????????? 7 ????? Parabola ??????????? -
?????????????????? Y ??????????? Saddle
?????????? 7 ????? Parabola ?????????
66
Saddle Point
??????????? x 0 ???????? Parabola
?????????????????? (minimum) ??? (0,0,0)
?????????????????? Minimax ???? Saddle Point
(????????????? Thomas Calculus)
??????????? x 0 ???????? Parabola
?????????????????? (Maximum) ??? (0,0,0)
67
Vector-Valued Functions in 3-D Space Curve
Space curve ??????????????? curve ?? 3
?????????????????? 3 components ??? P(x,y,z) ??
curve?????????
I time interval
?????????????????? Space curve
??? ??????????????????? ?????????????????
(????????????? Thomas Calculus)
Space curve ?????????? Position vector ???????
68
Examples Space Curves
Function ???????? vector-valued function
???? vector function
(????????????? Thomas Calculus)
69
Example Helix
Helix ??? space curve ????????????????????????????
?????? Helix ????? ??????????????????
???????????????????? x ??? y ????????
????? 2 ???????? (x,y) ???????????????????
???????????????? x ??? y ??????????????????? ?????
??????????? z ???????? Helix ?????? ?????? t
??????????????????????????????
????????????? Helix ???????????? Cylinder
(????????????? Thomas Calculus)
70
Examples Helix
?????????????????? z ???????????????????????
????????????????? z ???? ?????????????????? Helix
??????????????????????
(????????????? Thomas Calculus)
71
Examples Helix
MATLAB Code
??????????????????? t0 ?????? t ????????????
0.1 ????? ?????????? t 30
t 00.130 x cos(t) ysin(t) z
t plot3(x,y,z)
???????? x,y,z
??????
72
Limit
???????? Limit ?????? vector functions
(????? limit ????? component ??? ???????)
Continuity
A vector function ???????????
(continuous) ?????? t t0 ??????????
?????? continuous function ??????????
??????????????????? ?? Domain ???
73
Example Limit and Continuity
????????????????
74
Derivative at a Point
?????
(????????????? Thomas Calculus)
A vector function
is differentiable at t
t0 if f, g and h are differentiable at t0.
is differentiable if it is
differentiable at every point of its domain.
75
Tangent Line
??????????????????? ?????? P0(x0,y0,z0)
???????????????????? P0 ??????????? ???????
Tangent Line
P0
0
76
Smooth Curve
Curve traced by is smooth if
is continuous and
Piecewise Smooth
Curve ????????????? smooth curve ????? curve
???????? ??????????? Piecewise smooth curve
77
Velocity, Speed, Acceleration, Direction of Motion
????? ?????? ?????????????????????????
????????? smooth curve
1. ???????? (Velocity)
??????????? curve
2. ????????? (Speed) ??????????????????????
3. ???????? (Acceleration)
4. ??????????????????? (Direction of Motion) ???
78
Example A Hang Glider
??????????????????????????????????????????????????
???????
????????? Velocity, Speed, Acceleration
??????????????????? Velocity ??? Acceleration
??????????
79
Differentiation Rules for Vector Functions
??? ??? ???? Differentiable
vector functions ????????? t
??????????????? vector, c ??????????????? scalar,
??? f ???? differentiable scalar function
1. Constant Function Rule
2. Scalar Multiple Rules
3. Sum and Difference Rules
80
Differentiation Rules for Vector Functions
4. Dot Product Rule
5. Cross Product Rule
6. Chain Rule
81
Proof of the Cross Product Rule
82
Proof of the Chain Rule
???
83
Derivative of Triple Scalar Product
Show that if are
differentiable vector functions of t, then
84
Vector Functions of Constant Length
?????????????????????? ??????????????????????
?????????????????????????????? Origin Position
vector ??????????????????????????????
??????????????????????? ??????????? ???????????
Vector function of Constant length
??????????????????
?????? ??? Curve ???? ??? Curve
????????? ????????? ?????????????? ??????????
??????????
????
(????????????? Thomas Calculus)
85
Property of Vector Functions of Constant Length
????????????
Take derivative ???????????
????????
86
Integral of Vector Function
A differentiable vector function is an
antiderivative of a vector function
on an interval I if at each point of I.
Indefinite Integral
????????
87
Definite Integral
If the component of
are Integrable over
a,b, then the definite integral of From a to b
is
Example
88
Example the Flight of a Glider
An acceleration of a glider is
At time t 0, the glider is at (3,0,0) with
velocity
Find the equation of the glider position
???
???
??????? ????????????
??????????
???
???
??????? ????????????
??????????
89
Example Motion along a Cycloid
Curve
is called a cycloid.
Find the maximum and Minimum of and
90
Arc Length Along a Curve
?????????? smooth curve ???????????
or
91
Example Distance Traveled by a Glider
?????????????????????????????????????????????
???????????????????????????????????????????? t
0 ??? t 2p
(????????????? Thomas Calculus)
92
Arc Length As a Function of Time
?????????????? Arc length ??????? function of t
??????
????? ???? Base point
????????????????? s(t) ???????????????????????????
?????????????? ????????????????? t
??????????
????????
????????
93
Arc Length Parameterization
???????????
???????????????????????????????????? ??????
???? curve
?????????????? inverse function ??? s(t)
??? ????????
???????????????? ????????????
???
???????????????? ?????????????????????????
? s ??????????????
???????????????? ???????????????? s ???
Arc length parameterization
94
Arc Length Parameterization
??????????????????????????????????????????
??????????????????? ???? ?????????????????????????
????????????????????????????????????????????? ????
???? 90 km/hr. ???????????????????????????????????
????????????????? ??????????????????
???????????????????????????????????????????
???????????? ????????????????????????????????????
??????????????????? 30 km ?????????????????
???????????????????????? ????????????
??????????? ????????????????????????????
???????????????????????? Arc length
parameterization
95
Length is independent of Parameterization
??????? Arc length parameterization ????
function ?????????????????? parameter
?????????????? parameter ?????????????
???
???
???????? ????????????????????? 1 ?????? 2
??????????????????????????????????? ??????????????
??????????????????????????? 2
96
Speed on a Smooth Curve
Speed ????????????????????????
???????????? ???????????????????????? ????????????
?????? s ???????????? t ????????
Unit Tangent Vector
??????????????????????????????
(????????????? Thomas Calculus)
?????????
???
???????
97
Unit Tangent Vector
?????
Example
98
Curvature
Curvature ????????????????? curve
????????????????????????????? ????????????????????
???????????????????????? curve ????????
?????
If is the unit tangent vector of a
smooth curve, the curvature function of a smooth
curve is
?????????????????
99
Example Curvature of a Circle
?????????? a ???????????????
100
Principal Unit Normal
??????? ????????????
????????? ???????????
????? At any point where the
principal unit normal vector for a curve in the
plane is
????????????????????
(????????????? Thomas Calculus)
101
Example Principal Unit Normal
102
Circle of Curvature
Definition The circle of curvature or osculating
circle at a point P on a plane curve where
is the circle in the plane of the curve
that 1. is tangent to the curve at P 2. has the
same curvature the curve has at P 3. lies toward
the concave or inner side of the curve
Radius of Curvature
Center of Curvature center of the circle
(????????????? Thomas Calculus)
103
Example Find Osculating Circle of a Parabola at
t 0
(????????????? Thomas Calculus)
104
Tangential and Normal Components of Acceleration
???
???????
(????????????? Thomas Calculus)
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