vector calculus: are you ready? vectors in 2d and 3d...
TRANSCRIPT
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Vector Calculus: Are you ready?
Purpose: Make certain that you can define, and use in context, vector terms,
concepts and formulas listed below:
Section 7.1-7.2
find the vector defined by two points and determine the norm of the vector.add two vectors multiply a non-zero vector by a non-zero scalar.represent a non-zero vector in the xy-plane in terms of its magnitude and the angle it makes with the positive x-axis.
Vectors in 2D and 3D Space: Review
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Vectors in 2D space
There are many quantities that are vector functions: Some Daily Use of Vectors
A wind of 80 km/h from the Southeast.A car going 80 km/h East.A vertical velocity of 20 m/s.A plane traveling 1000 km/h on a 180 heading.
These issues are described by a magnitude and a direction.
Vector algebra and vector calculus have resulted from practical engineering applications: Mechanics, Fluid flows, Wireless CommunicationsScalar: is described by a single quantity such as work, energy, potential, speed, temperature, blood pressure ..Vector: is described by a magnitude and direction such as velocity, electric force, position of a robot …
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Some Applications of VectorsMechanics: Force, Torque, position, speed, acceleration, …Electromagnetism: Electric and magnetic fields, current density, pointing vector,…Example Walking and Different Forces
Example Mechanical System in Equilibrium
Other Examples of vector quantities
Notation ur,v⎯→⎯
AB
Acknowledgment: Most figures included in class notes are copied from the textbook by Zill and Cullen.
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Notation and TerminologyA vector with starting point A and end point B is written asMagnitude of is written as:
⎯→⎯
AB
⎯→⎯
AB||||
⎯→⎯
AB
Two vectors with the same magnitude and direction are equal
Parallel vectors: nonzero scalar multiples of each other
Example: In 2D Cartesian Coord.:
22
21
212121
:Magnitude
],[ , ˆˆ
aaa
aaaajaiaa
+=
=><=+=r
r
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A note about notation
The textbook uses boldface to represent vectors,I may place an arrow above general vectors and a hat over unit vectors. I would like you all to clearly identify vectors in your work.
Fr
F u u ii ˆ=
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Addition of Vectors
The sum of two vectors is the main diagonal of the parallelogram with the vectors as sides
⎯→⎯⎯→⎯⎯→⎯
+= ACABAD
Consider two vectors and with common initial point A
⎯→⎯
AB⎯→⎯
AC
Example:
ijijijijijiji
100106664210)66()44(
=−=−++
−=−++
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Subtraction
Subtraction: The difference of and is defined by
)(⎯→⎯⎯→⎯⎯→⎯⎯→⎯
−+=− ACABACAB
⎯→⎯
AB⎯→⎯
AC
⎯→⎯
AB⎯→⎯
− AC
⎯→⎯⎯→⎯
− ACAB is the main diagonal of the parallelogram with sides and
⎯→⎯⎯→⎯⎯→⎯
−= ACABCBOr is a vector from the end of the second vector toward the end of the first vector
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At equilibrium: F1 + F2 + w = 0
Sphere weight=50 lb
2 supporting planes
w = -50 j lb
∴ F1 = 25.9 lb, F2 = 36.6 lb
Review Exercise (page 346): Prob. 48Find the magnitude of F1 and F2.
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Properties of VectorsMagnitude, length, or norm of a vector a: ||a||
If then:>=< 21 aa ,a22
21 aa +=|||| a
1||||1ˆ =⎟
⎠⎞⎜
⎝⎛= u with aau
ji 212121 ,00, , aaaaaa +=><+><=><The i, j unit vectors: i=<1,0>, j=<0,1>
Example: Given a=<3,-4>, form a unit vector • in the same direction as a. Answer: <0.6,-0.8> • In the opposite direction of a. Answer: <-0.6,0.8>
A vector that has magnitude 1 is called unit vector.A unit vector in the direction of a is:
ij u
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7.2 Vectors in 3-SpaceRectangular or Cartesian Coordinate2D-Space: Two orthogonal axes
The three axes follow the Right Hand Rule
: Three mutually orthogonal axesSpace-3D
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Coordinate Plane: Each pair of coordinate axes determines a coordinate plane (xy,xz and yz).Octant:The coordinate planes divide the 3-space into 8 parts known as Octants.First octant: x, y, z>0
3D-Space
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P 1P 2
Given 2 points:
>=<= 1111 ,, zyxPOrrr
Position Vector:For a point P, the position vector is
>−−−=<−=
121212
1221
,, zzyyxxPOPOPPrrr
Vector between two points:
),,(P & ),,(P 22221111 zyxzyx
212
212
212 zzyyxxd )()()()P,P( 21 −+−+−=
Distance Formula between two points:
Examples: P1 = (1,2,3) & P2 = (1,-1,-1)
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Component Definitions in 3D-SpaceLet and be vectors in
(i) Addition:
(ii) Scalar Multiplication:
(iii) Equality: if an only if
(iv) Negative of a vector:
(v) Subtraction:
(vi) Zero vector: 0 = <0,0,0>
(vii) Magnitude:
>=< 321 aaa ,,a >=< 321 bbb ,,b 3R>+++=<+ 332211 bababa ,,ba
>=< 321 kakakak ,,aba= 332211 bababa === ,,
>−−−=<− 321 bbb ,,b>−−−=<−+=− 332211 bababa ,,)( baba
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22
21 aaa ++=|||| a
kjia 321321 aaaaaa ++>==< ,,
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Unit Vectors in 3D space:
i = <1,0,0>,
j = <0,1,0>,
k = <0,0,1>
kji a 321321 ,, aaaaaa ++=><=
Example: If a = 3i - 4j + 8k and b = I - 4k, find 5a - 2b.
b = i- 0j - 4k 2b = 2i + 0j - 8k
5a = 15i - 20j + 40k
5a - 2b = 13i - 20j + 48k
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2. Section 7.3define the dot (inner) product (a . b) and interpret it geometrically.use the dot product to determine: – work done by a force, – the angle between two vectors, – whether two vectors are perpendicular to one
another, – projections and components of vectors, – direction angles and direction cosines
7.3 Dot (scalar or inner) Product
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Applications: Mechanics and Electromagnetism
πθθ ≤≤⋅=⋅ 0 & cos|||||||| baba
Definition:
The dot product of two vectors a and b
is the scalar
θ is the angle between a and b
Example Dot producti . i=1, j . j=1, k . k=1 since ||i||=||j||=||k||=1 and θ = 0i . j=0, j . k=0, k . i=0 since θ = 90o
Example Given:a=10i+2j-6k, b=-0.5i+4j-3k a . b=(10)(-0.5)+(2)(4)+(-6)(-3)=21
Dot (scalar or inner) Product
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Physical Interpretation of the Dot ProductA constant force of magnitude F moves an object a distance d in the direction force, the work done by the force (W):
θθcos||||||||
||||)cos||(||d F
dF ==
⋅= dFWrr
|||||||| d F=⋅= dFWrr
When a constant force F applied to a body acts at an angle θ to the direction of motion, the work done by the force (W):
Note: if F and d are orthogonal, W=0.
Examples:
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Work done by w (gravity force) = w . d = 0 (w ¦d)
Work done by F (applied force) = F . d = |F|.|d| cos θ (d // F)= 150 N.m
md
NF
>=<
=
3,4
30||||r
P7.3-47: Given
weight
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Properties of Dot Producta . b = 0 if a=0 or b=0 a . b = b . a (commutative law)a . (b+c) = a . b+a . c (distributive law)a . (kb) = (ka) . b = k(a . b) k a scalara . a ≥ 0a . a = ||a||2
For nonzero vectors a and b(i) a . b > 0 if and only if θ is acute(ii) a . b < 0 if and only if θ is obtuse, and(iii) a . b = 0 if and only of if cos θ =0 (Orthogonal vectors)
Theorem 7.1 Criterion for Orthogonal VectorsTwo nonzero vectors a and b are orthogonal if and only if a . b=0
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Angle Between Two Vectors:
||||||||||||||||cos 332211
b ab ababababa ++
=•
=rr
θ
27214cos =θ o9.4477.0
942cos 1 ≈≈⎟⎟
⎠
⎞⎜⎜⎝
⎛=∴ −θ
ExampleFind the angle between a = 2i+3j+k & b = -i+5j+k.
27||||,14|||| == baa . b=14,
Solution:
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For a nonzero vector in 3D-Space the angles α, β and γ with i, j, and k are called direction angles of a.
Direction cosines for
ooo 4.53,8.41,7.72 ≈≈≈⇒ γβα
kjia 321 aaa ++=
||||||ˆ||||||ˆ
cos 1
a aa
iia
=•
=r
α
||||cos 2
aa
=β||||
γcosa
3a=
1coscoscos 222 =++ γβα
Example Find the direction cosines and direction angles of the vector a = 2i+5j+4k. ||a||=6.71
Direction Angles
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Component of a on b:
( ) vector
ˆ ˆ||||
)comp(proj
=
⋅=⎟⎟⎠
⎞⎜⎝
⎛= bba
bbaa bb
r
scalar
ˆ||||
cos||||comp
=
⋅=⋅
== bab
baaab θ
Projection of a in the direction of b:
Example: a = < -1,-2,7 > & b = < 6,-3,-2 >
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3. Section 7.4
define the vector (cross) product (a x b) and interpret it geometrically.determine the cross product of vectors and combinations of vectors, use to determine torquefind unit vectors that are perpendicular to two given vectors.
7.4 Cross (Vector) Product
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Cross (Vector) ProductThe vector product of 2 vectors A and B is given by
amparalleogr of area ||BA|| BA
B B BA A Ak j i
BA
zyx
zyx
=×
∗∗=
−−−==×
n
BABAjBABAi xzzxyzzy
ˆ)sin(
)(ˆ)(ˆˆˆˆ
θ
)(ˆ xyyx BABAk −+
where n is a unit vector perpendicular to A and B, pointing in the direction given by the right hand screw rule (i.e. the direction in which a screw would advance if it were turned from A to B.
A
Bθ
AxB
n
Example: a = < -1,-2,7 >
& b = < 6,-3,-2 >
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Typical Applications
b) AREA OF A Triangle with edges a and b:
Area = 1/2⎪⎢a × b ⎪⎢= 1/2 ⎪⎢a⎪⎢⎪⎢b⎪⎢ sin θ
a) AREA OF A PARALLELOGRAM with edges a and b:
Area = ⎪⎢a × b ⎪⎢= ⎪⎢a⎪⎢⎪⎢b⎪⎢ sin θ
Example (p7.4, # 48): Find area of the triangle through:
p1 = (0,0,0), p2 = (0,1,2), P3 = (2,2,0)
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Typical Applications
Volume of a parallelepiped (with edges: a, b & c)
Volume = (area of base) . (height)
= ⎪⎢b × c ⎪⎢ .⎮comp b × c a⎮
= ⎪⎢b × c ⎪⎢. ⎮a • (b × c) ⎮ / ⎪⎢b × c ⎪⎢
∴Volume =⎮a • (b × c)⎮
Example: a = < 3,1,1 >,
b = < 1,4,1 >
& c = < 1,1,5 >
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Typical applicationsMOMENT OF A FORCE
In mechanics the moment m of a force F about a point Q is defined as the product
m =⎪⎢F ⎪⎢ d
d = ⎪⎢r ⎢⎢ sin θ
rFdrFm rrr×=∗=∴ ˆ sin θ
m is called the moment vector or vector moment of F about Q
where d is the (perpendicular) distance between Q and the line of action L on F.
Q d=r sin θ
If r is the vector from Q to any point Aon L, then
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• Torque T = r x F
• Force on a moving charge due to a magnetic field due to
FB
vqF = q v x B
• Velocity of a rotating body
v = w x rω= angular speed
|w| = ω and directed along axis of rotation
w
r
v
ω
o
More Applications
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Two non-zero vectors A and B are parallel if & if : 0BA =×rr
jkiijkkij
jikikjkjiˆˆˆ ˆˆˆ ˆˆˆ
ˆˆˆ ˆˆˆ ˆˆˆ
−=×−=×−=×
=×=×=×
• Circular Mnemonic:
• More Cross Product Properties:
Cross product is not commutative:
ABBA
ABBArrrr
rrrr
×−=×
×≠×
Cross product is not associative: C)BA()CB(Arrrrrr
××≠××Example (p7.4, # 13): A = <2,7,-4>, B = <1,1,-1>
Find a vector that is perpendicular to A and B
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Purpose: Make certain that you can define, and use in context, vector
terms, concepts and formulas listed below:
4. Section 7.5express a line as a: vector parameterization, and scalar parameterization,use vectors to determine whether two lines intersect, and if so, the point of intersection.use vectors to find the distance from a point to a line.express a plane as a scalar equation and as a vector equation.find whether two planes intersect, and if so, the angle of intersection and a vector parameterization of the line formed by the intersection.unit normal for a plane.
7.5 Lines and Planes
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Equation of a “straight” Line
>=<>=<
22222
11111
,,:P& ,,:P
zyxrzyxr
r
r
aatrrtrrtrr
rrrr
rrrr
of direction thein is line the,parameterscalar ),(
2
122
+==−+=
12
2
12
2
12
2
zzzz
yyyy
xxxx
−−
=−−
=−−
1. Vector equation of the line through r1 & r2:
2. Parametric & symmetric equations of the line:
Given two points in 3D:
Two forms for the line through P1 & P2:
If a is a unit vector, then its components are direction cosines of the line.
Examples to follow:
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Examples: 7.5, # 3
parameter scalar =−+= trrtrr ),( 211rrrr
Find the vector equation of a line through: (1/2, -1/2, 1) & (-3/2, 5/2, -1/2).
Examples: 7.5, # 27
Show that the two lines:
r = t <1,1,1> and r = <6,6,6> + t <-3,-3,-3>
are the same.
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7.5 Equation of a Plane
kcjbian ˆ ˆ ˆ ++=r
0 z y x =+++ dcba
1. The equation of a plane perpendicular to a normal vector
is given by:
2. The equation of a plane contains 3 points: P1(r1), P2(r2), P3(r3) is given by:
[ ] 0)()()( 11312 =−•−×− rrrrrr rrrrrr
Examples to follow:
Two forms:
a vector form.
0( =• n)r-r 1rrr
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Examples: 7.5, # 39
0 z y x =+++ dcba
Examples: 7.5, # 51
Find the equation of a plane contains: (5,1,3) & perpendicular to <2,-3,4>
< r-r1 > . n = 0
Two methods:
Find the equation of a plane contains: (2,3,-5) & parallel to x + y - 4z = 1
Or:
Answer:
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Intersection of Two PlanesLeta1 x + b1 y + c1 z = d1 &a2 x + b2 y + c2 z = d2
be two non parallel planes.
We get a system of two equations and three unknowns.
Choose one variable arbitrary, say x = t, and solve the new system of two equations and two unknowns y and z.
parametric equations for the line of intersection
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Example Find the parametric equation for the line of intersection of
2x – 3y + 4z = 1 and x – y – z = 5
SolutionLet choose z = t, sub in the 2 equatinsand solve for x and y from
2x – 3y = 1 – 4t and x – y = 5 + t
Then, x = 14 + 7t, y = 9 + 6t, z = t
END of selected materials from Chapter 7.