module m2-1 2 electrical engineering -...
TRANSCRIPT
T U T O R I A L 1 V E C T O R A N A L Y S I S A U G U S T 4 , 2 0 1 6
Module M2-1 Electrical Engineering
1
Topics
� Review of vectors (with Matlab) � Examples � In-class exercises
2
After this tutorial, you will be able to 3
� understand the concepts of the unit vector, rectangular coordinate system, and vector field
� add vectors, subtract vectors, and multiple a vector by a scale
� compute the magnitude of a vector, the dot product, and the cross product
� write a simple Matlab program to compute vector algebra
Notation
Vector or or
Scalar
Vector and scalar quantities
� Vector is a quantity that has both magnitude and direction
� We usually represent a vector by a bold font, or a font with an arrow or a hat on top
A
A
4
~A bA
Use an arrow for an arbitrary vector
Use a hat to emphasize that a vector has length (magnitude) 1
In this class, we will use these 2 notations
The Cartesian coordinate system
� The direction of the axes is determined by the right-hand rule
� Each point in the space is represented by a triple (x, y, z)
x
y
z
x
y
z
5
The vector from one point to another point
� The vector from point to point is given by or
http://www.colorado.edu/geography/ gcraft/notes/coordsys/coordsys_f.html
(x2, y2, z2)(x1, y1, z1)
6
~A = (x2 � x1)bi+ (y2 � y1)bj+ (z2 � z1)bk
~A =⇥x2 � x1 y2 � y1 z2 � z1
⇤
Concept question 7
ข้อใดคือเวกเตอร์ในสองมิต ิที่เริ่มต้นจากจุด (-2,1) และสิ้นสุด ที่จุด (3,-5)
ก.
ข.
ค. ไม่มีข้อใดถูก
(ส่วนหนึ่งของข้อสอบปลายภาคปีการศึกษา 1/2556)
5bi� 6bj
6bi� 5bj
Vector operation in Matlab
� Matlab represents a vector by >> A = [1 2 3]
� For a vector � The magnitude (or length) of is given by
� Matlab command for the magnitude >> norm(A)
8
~A =bi+ 2bj+ 3bk
~A =⇥A
x
Ay
Az
⇤= A
x
bi+Ay
bj+Az
bk
|~A| =qA2
x
+A2y
+A2z
~A
A unit vector 9
� A unit vector is a vector of magnitude 1
� Example: is a unit vector because
~M =1p6bi+ 2p
6bj� 1p
6bk
| ~M| =
s✓1p6
◆2
+
✓2p6
◆2
+
✓� 1p
6
◆2
=
r1
6+
4
6+
1
6
= 1
A unit vector in the direction of vector .
10
� is the vector that points to the same direction as and has the magnitude of 1
~B
~B
length 1
baBA unit vector in the direction of vector . ~B
~B
Example 1
baB
A unit vector in the direction of vector .
~B
length 1
~B
Example 2
(continued) 11
� In general for vector , the unit vector in the direction of is
� Example: The unit vector in the direction of is
~B =⇥B
x
By
Bz
⇤= B
x
bi+By
bj+Bz
bk~B
baB
=~Bq
B2x
+B2y
+B2z
=~B
|~B|
x
y
-3
0
-1
~B = �3bj
~B = �3bj
baB =~B
|~B|=
�3bj3
= �bj
baB = �bj
Dot product
� For vector and vector
� The dot product is a scalar that equals
� Matlab command >> dot(A,B)
12
~A =⇥A
x
Ay
Az
⇤= A
x
bi+Ay
bj+Az
bk~B =
⇥B
x
By
Bz
⇤= B
x
bi+By
bj+Bz
bk~A • ~B
~A • ~B = Ax
Bx
+Ay
By
+Az
Bz
= |~A||~B| cos ✓
~A ~Bthe angle between and
Cross product
� The cross product is a vector that equals
� Matlab command >> cross(A,B)
13
~A⇥ ~B
~A⇥ ~B =
������
bi bj bkA
x
Ay
Az
Bx
By
Bz
������
= (Ay
Bz
�Az
By
)bi+ (Az
Bx
�Ax
Bz
)bj+ (Ax
By
�Ay
Bx
)bk
determinant
The determinant of a 3-by-3 matrix is easy to compute by hand
14
� Multiply the individual items � Put the negative signs for the upward terms � Combine the terms
������
bi bj bk1 2 34 5 6
������
�(4)(2)bk�(5)(3)bi�(6)(1)bj
(2)(6)bi
(3)(4)bj
bi bj1 24 5
=
������
bi bj bk1 2 34 5 6
������(1)(5)bk
= �8bk� 15bi� 6bj +5bk+ 12bj+ 12bi
= �3bi+ 6bj� 3bk
Magnitude and direction of .
� The magnitude is � The direction is orthogonal to both and and in
the direction of the thumb (right-hand rule)
✓
15
~A⇥ ~B
~A⇥ ~B
~A
~B
|~A⇥ ~B| = |~A||~B| sin ✓
~A ~B
~A ~Bthe angle between and
right hand
Example: cross product
The cross product of the two vectors and is shown in blue
16
~A = 2bi+bj+ 0bk~B = 1bi+ 2bj+ 0bk
~C = 0bi+ 0bj+ 3bk
Concept question 17
มีขนาดและทิศใด เมื่อ และ
ก. ขนาด 1 ชี้ตามแกน -z
ข. ขนาด 2 ชี้ตามแกน +z
ค. ไม่มีข้อใดถูก
(ส่วนหนึ่งของข้อสอบปลายภาคปีการศึกษา 1/2556)
~A⇥ ~B ~B =⇥0 �1 0
⇤~A =
⇥2 0 0
⇤
Example: 2D Cartesian coordinate system
� Let vectors , , and equal
� Question 1: Draw these vectors in the same diagram
18
~A ~B
~A =⇥3 �2
⇤ ~C =⇥�2 0
⇤~B =
⇥1 3
⇤~C
Solution (picture) to question 1 19
−5 −4 −3 −2 −1 0 1 2 3 4 5−5
−4
−3
−2
−1
0
1
2
3
4
5
x
y
~A =⇥3 �2
⇤
~C =⇥�2 0
⇤
~B =⇥1 3
⇤
Example: 2D (cont.)
� Question 2: Find in Matlab and draw in paper the vector sums and
� Solution (Matlab): >> clear all % clear all variables >> A = [ 3 -2]; >> B = [ 1 3]; >> C = [-2 0];
20
~A+ ~B ~A+ ~B+ ~C
>> A+B ans = 4 1 >> A+B+C ans = 2 1
Solution (picture): .
triangular law parallelogram law
21
~A+ ~B =⇥4 1
⇤
−4 −3 −2 −1 0 1 2 3 4−4
−3
−2
−1
0
1
2
3
4
x
y
~A
~B
~A+ ~B
−4 −3 −2 −1 0 1 2 3 4−4
−3
−2
−1
0
1
2
3
4
x
y
~A
~B~A+ ~B
Solution (picture): .
triangular law parallelogram law
22
~A+ ~B+ ~C =⇥2 1
⇤
−4 −3 −2 −1 0 1 2 3 4−4
−3
−2
−1
0
1
2
3
4
x
y ~A+ ~B
~A+ ~B+ ~C
~C
−4 −3 −2 −1 0 1 2 3 4−4
−3
−2
−1
0
1
2
3
4
x
y ~A+ ~B
~C~A+ ~B+ ~C
Example: 2D (cont.)
� Question 3: Find and draw vector differences , , and
� Solution (Matlab): >> A-B ans = 2 -5 >> B-A ans = -2 5 >> A-B-C ans = 4 -5
23
~A� ~B� ~C~A� ~B ~B� ~A
Solution (picture) 24
−6 −4 −2 0 2 4 6
−6
−4
−2
0
2
4
6
x
y
~B� ~A
~A� ~B~A� ~B� ~C
Example: 2D (cont.)
� Question 4: Find and draw scalar multiplications of vectors , , and
� Solution (Matlab): >> 2*A ans = 6 -4 >> -3*B ans = -3 -9 >> C/1.5 ans =
-1.3333 0
25
2~A �3~A ~C/1.5
Solution (picture): .
26
2~A =⇥6 �4
⇤
−8 −6 −4 −2 0 2 4 6 8−8
−6
−4
−2
0
2
4
6
8
x
y
2~A
~A
Solution (picture): .
27
�3~B =⇥�3 �9
⇤
−10 −8 −6 −4 −2 0 2 4 6 8 10
−10
−8
−6
−4
−2
0
2
4
6
8
10
x
y
�3~B
~B
Solution (picture): .
28
~C
1.5=
⇥� 4
3 0⇤
−3 −2 −1 0 1 2 3−3
−2
−1
0
1
2
3
xy
~C
~C
1.5
Example: 2D (cont.)
� Question 5: Find the dot product:
� Solution: >> dot(A, B) ans = -3
29
~A • ~B
Example: 2D (cont.)
� Question 6: Find and draw the cross product
� Solution: >> cross([A 0], [B 0]) ans =
0 0 11
Append zero to the z-component, so we have a 3D vector needed for computing a cross product
30
~A⇥ ~B
Solution (picture) 31
−10−5
05
10
−10−5
05
10
−10
−5
0
5
10
xy
z
~A⇥ ~B
~A ~B
Vector field is a type of functions 32
� Definition: A vector field in two dimensions is a function that assign to each point a two-dimensional vector
� To picture a vector field, we draw the arrow representing the vector starting at the point
(x, y)
(x, y)
. (x, y)
~F(x, y)
~F(x, y)
~F(x, y)
~F
ปริมาณเวกเตอร์ที่พิกัดนั้น
ตำแหน่ง หรือพิกัด
Example of a vector field in 2D 33
� Example: A vector field in 2D is defined by Describe the field by sketching some of the vectors
� Solution: see next page
~F(x, y) = �y
bi+ x
bj
~F(x, y)
34
Given that , so
0
y
x
1 2
3 continuing this way, we draw a number of representative vectors
3
~F(x, y) = �y
bi+ x
bj1 ~F(1, 0) = �(0)bi+ (1)bj = bj
2 ~F(0, 1) = �(1)bi+ (0)bj = �bi
A vector field in 3D 35
� Definition: A vector field in three dimensions is a function that assign to each point a three-dimensional vector
(x, y, z)
~F(x, y, z)
~F
ปริมาณเวกเตอร์ที่พิกัดนั้น
ตำแหน่ง หรือพิกัด
Example of a vector field in 3D 36
� Example: A vector field is given by Evaluate at position (2, 4, 3)
� Answer:
~S
~S(x, y, z) =4
(x� 1)2 + (y � 2)2 + (z + 1)2
h(x� 1)bi+ (y � 2)bj+ (z + 1)bk
i
~S
~S(2, 4, 3) =4n
(2� 1)bi+ (4� 2)bj+ (3 + 1)bko
(2� 1)2 + (4� 2)2 + (3 + 1)2
=4
21bi+ 8
21bj+ 16
21bk
= 0.19bi+ 0.38bj+ 0.76bk
put x = 2
y = 4
z = 3
37
(continued) Matlab plot of is below
−100
10
−10
0
10
−10
0
10
xy
z
~S(x, y, z) Summary 38
� Vector arithmetic ¡ vectors in Matlab ¡ magnitude, unit vector ¡ sum, difference, scalar multiplication ¡ dot product, cross product ¡ vector field
39
In-class exercise
Homework 1(a) 40
Given the vectors and , find the magnitude of
~M =bi� 2bj+ 3bk ~N = 2bi� bkbk� 2~N+ ~M