Title: Vectors and Motion In Space
1Vectors and Motion In Space
Chapter 10
- 170 121 Engineering Mathematics II
- 16 ????????? 2547
2Content
?????????????? Cartesian Coordinate, vector ???
operations ?????????, ????????????????????????, ??
????????????, vector-valued functions ??? curve
???????, ?????????? curve ??? unit vector
???????????? curve ????? Space
(???????????? ?????) ?????????????????????????????
??? ????? ????? ?????????? ???????????????????????
???? ????????????????????? Three Dimensional
Space ?????????????????????????????????????????
????????????????? (???????? Space
?????????????????????????? ???????????????????????
????????? ???????????????? ???????????????????????
?????????????????????????????????????????? ???????
?????? Space ????????????????)
3Cartesian 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
4Cartesian 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
5Octants
?????????????? 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
6Cylinder
?? 3 ????????????????????????????????????? ????
???? Z ??????????????????????
??? R ????????????????????
??????????????????????????? ????????????????? ???
z ?????? ????????????????? ?????? z
??????????????? ????????????????? z
??????????? ?????????? ????? 1 ????
??????????????? ??????? z ?????????????????? ?????
?????????
(????????????? Thomas Calculus)
7Component Form of a Vector in Space
??????????????? vector ?? Space 3
????????????????? ???? ??? Component Form
Z
v1, v2, v3 ???????????????? ????????
X,Y, Z ????????
Y
X
8Standard Unit Vectors For 3D Space
??????????????? vector ?????? standard unit
vector ??????
??? ??? standard Unit
vectors ????????? X, Y, Z ???????? ??????
Z
Y
X
9Position Vector
??????????? vector ??????????????? ????????
vector ?????? Position vector
????
???? Position vector ?????? P(x,y,z)
(????????????? Thomas Calculus)
10Directed Line Segment
(????????????? Thomas Calculus)
11Magnitude of Vectors
??????? vector ?????????????????????
(????????????? Thomas Calculus)
12Unit Vector
???????????????????? vector ?????????? unit
vector ?????? 1 ????
?????????????? vector ????????
????????????? ?????????
???????
?????????
13Dot Product
?????
???? operation ??????? vector ??? vector
???????????? Scalar ????????????????????? vector
?????????
??? ?????????? ?????
14Properties of Dot Product
1.
2.
3.
4.
5.
15Vector Projection
Projection of onto
???????????????????? ????????????????
(????????????? Thomas Calculus)
16Vector Project
????????????? ??????? 2 ??????????
????????????? ??????????
????????????? ?????????????
(????????????? Thomas Calculus)
17Cross Product
?????
???? Operation ??????? ???
???????????? vector ???????????????????????
???
(?????????????????)
(????????????? Thomas Calculus)
18Cross Product
???????
????????????????????????????????
(Parallelogram) ??????????????????? ???
(????????????? Thomas Calculus)
19Properties of Cross Product
1.
(????????????? Thomas Calculus)
20Properties of Cross Product
2.
3.
4.
5.
21Cross Products of Standard Unit Vectors
???
-
??
-
-
22Determinant Formula for Cross Product
23Application of Cross Product Torque
Torque ???????????????????????????????????????????
???? ???????????
(????????????? Thomas Calculus)
24Box Product (Triple Scalar Product)
????? Box product ???????
(?????????????) ????????
?????????????????????? parallelpiped ??????
(????????????? Thomas Calculus)
25Formula for Box Product
Box product ???????????????????????? Determinant
??????
???????????? Box product
26Lines in Space
???????????????????????? ???????????????????
P0 ????????????????? ????????? v ??????????
(????????????? Thomas Calculus)
???????????????????????????? P0 ??? P ???????????
vector ??????? v
????????
t ??????? scalar ????????
27Equations for a Line in Space
???
???
??????
???????????? Parametric Equations for a Line
??????
??????????????????????????????????????????????????
????
????????????? vector equation ????????
???
Position vector
28Example Lines in Space
????????????????????????????? (-2,0,4) ??? (0,4,2)
?????? 1. ??????????? 2. ??????????
0
(????????????? Thomas Calculus)
29Distance Between a Point and a Line
????????????? S ?????????? L ???????????
L
30Planes in Space
?????????????????????????????????????????????
??? P0 ????????????????? Vector
?????????????????? (Normal vector)
(????????????? Thomas Calculus)
????????? ??????????????? ??????? Dot
product ??????? vector ????? ????????
?????? 0 ????
31Equations for a Plane in Space
??? P0(x0,y0,z0) ??? P(x,y,z) ????????????????? M
????????
???? vector ??????? M ????????????????????????????
????
?????
???? normal vector ???????? M
32Equations for a Plane in Space
?????????????
??????????????????
?????
????
???
33Example a Plane in Space
?????????????????????????????? A(0,0,1), B(2,0,0)
??? C(0,3,0)
?????? 1. ????? vector 2 ????????????????????
2. ????? normal vector ???????? 3. ??????
34Example a Plane in Space
??????????????? A(0,0,1), B(2,0,0) ??? C(0,3,0)
(????????????? Thomas Calculus)
35Lines of Intersection Between Planes
?????????????????????? v ????????????????????????
???????????????????
(????????????? Thomas Calculus)
36Example a Line of Intersection Between Planes
??????????????????????????????????????????????????
1. ????????????????????? 2. ????????????????????
???? 3. ?????????????????
37Distance Between a point and a Plane
????????????? S ???????? M ???????????
q
M
(????????????? Thomas Calculus)
38Example Distance Between a point and a Plane
39Cylinders
?????????????????? Cylinder ??????????????
???????????????? ??????????? Cylinder
????????????????????????????????????????? ???
Cylinder ??????? ????????????????????????????????
?????????????????????????????????????
????????
????? ?? space 3
?????????????
???????
??????? ??? z ???? ?????? x0 ???????????
(????????????? Thomas Calculus)
40Example Cylinder
???????????????
(????????????? Thomas Calculus)
41Cylinders
???????????????????? - ??????????? Cylinder
????????? f(x,y) c ???????? Cylinder
????????? ???????????????????????? Z -
??????????? Cylinder ????????? f(x,z) c
???????? Cylinder ????????? ??????????????????????
?? Y ?????????? ???????? 2 ????????????
Cartesian coordinate 3 ???? ??????????? Cylinder
??????????????????????????????
???????????? Cylinder ????????????????????????????
???????????????? Cylinder ???????????????????
42Example Cylinder
?????????? Cylinder
(?????????? X)
(????????????? Thomas Calculus)
43Quadric 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 ????????
44Analysis 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 ??????????????????
45Ellipsoid
?????????????? 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)
46Cross Sections of an Ellipsoid
????? set ??? z z0 ??????
????? set ??? z 0 ??????????
????? set ??? x 0 ??????????
(????????????? Thomas Calculus)
47Cross 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)
48Elliptic Paraboloid
?????????????? Elliptic Paraboloid
??????????????? X ??? Y ???
(3)
??????????????? Origin (0,0,0) ???????????
Vertex ??? Graph ???????? z ?????????? Paraboloid
???????????????????????????????? ????????????
????????????????????????? Elliptic Paraboloid
???????? a b
(0,0,0)
(????????????? Thomas Calculus)
49Cross Sections of Paraboloid
????? set ??? y 0 ??????
????? set ??? z c ??????????
????? set ??? x 0 ??????
(????????????? Thomas Calculus)
50Cross Sections of Paraboloid
- Cross section ??????????????? z z0, z0 gt 0
??? Paraboloid ?????????? 3 ????????? Ellipse
???? - ??????????????????????? z ??????
??????????? Paraboloid ?????????? 3 ?????
Parabola ????
51Cone
?????????????? Elliptic Paraboloid ???????? z
?????????????? ???
(4)
??????????????? Origin (0,0,0) ?????????????
?? X ??? Y ?????? a ??? b ???????
???????????? ??????????? Circular cone
(????????????? Thomas Calculus)
52Cross 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)
53Cross Sections of a Cone
?????????????????? ????? set ??? z z0 ??? z0
?????????? 0, ??????????
????? set ??? x x0 ??? x0 ?????????? 0 ??????
????? set ??? y y0 ??? y0 ?????????? 0 ??????
54Cross Sections of a Cone
- Cross section ??????????????? z z0, z0
?????????? 0, ?????????????? ??? 4 ?????????
Ellipse ???? - ??????????????????????? z
?????? ????????????????????????? 4 ???????????? 2
??????????????????? (0,0,0) ????
55Hyperboloid
Hyperboloid ?? 2 ????
Hyperboloid of One Sheet
Hyperboloid of Two Sheets
(????????????? Thomas Calculus)
56Hyperboloid of One Sheet
?????????????? Hyperboloid of One Sheet
??????????????? Z ???
(5)
???????????????????????????????
???????????? Coordinate planes ??????? ??? a b
???????? Hyperboloid ?????? Surface of Revolution
(????????????? Thomas Calculus)
57Cross Sections of Hyperboloid of One Sheet
????? y 0 ??????
????? z c ??????
????? z 0 ??????
????? x 0 ??????
(????????????? Thomas Calculus)
58Cross Sections of Hyperboloid of One Sheet
- Cross section ??????????????? z z0 ???
Hyperboloid ??????? ??? 5 ????????? Ellipse ????
??? a b ?????????????????????????? -
??????????????????????? z ?????? ???????????
Hyperboloid ?????????? 5 ????? Hyperbola ????
59Hyperboloid 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)
60Cross Sections of Hyperboloid of Two Sheets
????? y 0 ??????
????? x 0 ??????
(????????????? Thomas Calculus)
61Cross Sections of Hyperboloid of Two Sheets
- Cross section ??????????????? z z0, z0 gt c,
??? Hyperboloid ?????????? 6 ????????? Ellipse
???? ??? a b ?????????????????????????? -
??????????????????????? z ?????? ???????????
Hyperboloid ?????????? 6 ????? Hyperbola ????
62Hyperboloids
Both hyperboloids are asymptotic to cone
??????????? 5 ??? 6 ????????????? ??? 1
????????????????????? 0 ????????????????????
??????????? Hyperboloid ???????????
(???????) ????????????
(????????????? Thomas Calculus)
63Hyperbolic Paraboloid a Saddle
???????????????? (Saddle) ???? Hyperbolic
Paraboloid ?????????????
(7)
??? c gt 0
Saddle ???????????? ????? x0 (?????
YZ) ???????? y 0 (????? XZ)
(????????????? Thomas Calculus)
64Cross Sections of a Saddle
????? x 0 ??????
????? z c ??????
????? z -c ??????
(????? y 0)
(????????????? Thomas Calculus)
65Cross 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 ?????????
66Saddle Point
??????????? x 0 ???????? Parabola
?????????????????? (minimum) ??? (0,0,0)
?????????????????? Minimax ???? Saddle Point
(????????????? Thomas Calculus)
??????????? x 0 ???????? Parabola
?????????????????? (Maximum) ??? (0,0,0)
67Vector-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 ???????
68Examples Space Curves
Function ???????? vector-valued function
???? vector function
(????????????? Thomas Calculus)
69Example Helix
Helix ??? space curve ????????????????????????????
?????? Helix ????? ??????????????????
???????????????????? x ??? y ????????
????? 2 ???????? (x,y) ???????????????????
???????????????? x ??? y ??????????????????? ?????
??????????? z ???????? Helix ?????? ?????? t
??????????????????????????????
????????????? Helix ???????????? Cylinder
(????????????? Thomas Calculus)
70Examples Helix
?????????????????? z ???????????????????????
????????????????? z ???? ?????????????????? Helix
??????????????????????
(????????????? Thomas Calculus)
71Examples 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
??????
72Limit
???????? Limit ?????? vector functions
(????? limit ????? component ??? ???????)
Continuity
A vector function ???????????
(continuous) ?????? t t0 ??????????
?????? continuous function ??????????
??????????????????? ?? Domain ???
73Example Limit and Continuity
????????????????
74Derivative 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.
75Tangent Line
??????????????????? ?????? P0(x0,y0,z0)
???????????????????? P0 ??????????? ???????
Tangent Line
P0
0
76Smooth Curve
Curve traced by is smooth if
is continuous and
Piecewise Smooth
Curve ????????????? smooth curve ????? curve
???????? ??????????? Piecewise smooth curve
77Velocity, Speed, Acceleration, Direction of Motion
????? ?????? ?????????????????????????
????????? smooth curve
1. ???????? (Velocity)
??????????? curve
2. ????????? (Speed) ??????????????????????
3. ???????? (Acceleration)
4. ??????????????????? (Direction of Motion) ???
78Example A Hang Glider
??????????????????????????????????????????????????
???????
????????? Velocity, Speed, Acceleration
??????????????????? Velocity ??? Acceleration
??????????
79Differentiation 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
80Differentiation Rules for Vector Functions
4. Dot Product Rule
5. Cross Product Rule
6. Chain Rule
81Proof of the Cross Product Rule
82Proof of the Chain Rule
???
83Derivative of Triple Scalar Product
Show that if are
differentiable vector functions of t, then
84Vector Functions of Constant Length
?????????????????????? ??????????????????????
?????????????????????????????? Origin Position
vector ??????????????????????????????
??????????????????????? ??????????? ???????????
Vector function of Constant length
??????????????????
?????? ??? Curve ???? ??? Curve
????????? ????????? ?????????????? ??????????
??????????
????
(????????????? Thomas Calculus)
85Property of Vector Functions of Constant Length
????????????
Take derivative ???????????
????????
86Integral 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
????????
87Definite Integral
If the component of
are Integrable over
a,b, then the definite integral of From a to b
is
Example
88Example 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
???
???
??????? ????????????
??????????
???
???
??????? ????????????
??????????
89Example Motion along a Cycloid
Curve
is called a cycloid.
Find the maximum and Minimum of and
90Arc Length Along a Curve
?????????? smooth curve ???????????
or
91Example Distance Traveled by a Glider
?????????????????????????????????????????????
???????????????????????????????????????????? t
0 ??? t 2p
(????????????? Thomas Calculus)
92Arc Length As a Function of Time
?????????????? Arc length ??????? function of t
??????
????? ???? Base point
????????????????? s(t) ???????????????????????????
?????????????? ????????????????? t
??????????
????????
????????
93Arc Length Parameterization
???????????
???????????????????????????????????? ??????
???? curve
?????????????? inverse function ??? s(t)
??? ????????
???????????????? ????????????
???
???????????????? ?????????????????????????
? s ??????????????
???????????????? ???????????????? s ???
Arc length parameterization
94Arc Length Parameterization
??????????????????????????????????????????
??????????????????? ???? ?????????????????????????
????????????????????????????????????????????? ????
???? 90 km/hr. ???????????????????????????????????
????????????????? ??????????????????
???????????????????????????????????????????
???????????? ????????????????????????????????????
??????????????????? 30 km ?????????????????
???????????????????????? ????????????
??????????? ????????????????????????????
???????????????????????? Arc length
parameterization
95Length is independent of Parameterization
??????? Arc length parameterization ????
function ?????????????????? parameter
?????????????? parameter ?????????????
???
???
???????? ????????????????????? 1 ?????? 2
??????????????????????????????????? ??????????????
??????????????????????????? 2
96Speed on a Smooth Curve
Speed ????????????????????????
???????????? ???????????????????????? ????????????
?????? s ???????????? t ????????
Unit Tangent Vector
??????????????????????????????
(????????????? Thomas Calculus)
?????????
???
???????
97Unit Tangent Vector
?????
Example
98Curvature
Curvature ????????????????? curve
????????????????????????????? ????????????????????
???????????????????????? curve ????????
?????
If is the unit tangent vector of a
smooth curve, the curvature function of a smooth
curve is
?????????????????
99Example Curvature of a Circle
?????????? a ???????????????
100Principal Unit Normal
??????? ????????????
????????? ???????????
????? At any point where the
principal unit normal vector for a curve in the
plane is
????????????????????
(????????????? Thomas Calculus)
101Example Principal Unit Normal
102Circle 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)
103Example Find Osculating Circle of a Parabola at
t 0
(????????????? Thomas Calculus)
104Tangential and Normal Components of Acceleration
???
???????
(????????????? Thomas Calculus)