Engr/Math/Physics 25
Chp2 MATLAB
Arrays: Part-1
Bruce Mayer, PE
Licensed Electrical & Mechanical Engineer
BMayer@ChabotCollege.edu
Engineering/Math/Physics 25: Computational Methods
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Bruce Mayer, PE
BMayer@ChabotCollege.edu • ENGR-25_Arrays-1.ppt
Learning Goals
 Learn to Construct 1-Dimensional
Row and Column Vectors
 Create MULTI-Dimensional
ARRAYS and MATRICES
 Perform Arithmetic Operations on
Vectors and Arrays/Matrices
 Analyze Polynomial Functions
Engineering/Math/Physics 25: Computational Methods
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Bruce Mayer, PE
BMayer@ChabotCollege.edu • ENGR-25_Arrays-1.ppt
Position Vector
 The vector p can be
specified by three
components: x, y, and z,
and can be written as:
p  xi  yj  zk or p  x, y , z 
• Where i, j, k are 
UNIT-Vectors Which are
|| To The CoOrd Axes
 MATLAB can accommodate Vectors
having more than 3 Elements
• See MATH6 for more info on “n-Space”
Engineering/Math/Physics 25: Computational Methods
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Bruce Mayer, PE
BMayer@ChabotCollege.edu • ENGR-25_Arrays-1.ppt
ROW & COLUMN Vectors
 To create a ROW
vector, separate the
elements by
COMMAS
>>p = [3,7,9]
p =
3
7
9
p  3, 7, 9
 The TRANSPOSE
Operation Swaps
Rows↔Columns
Engineering/Math/Physics 25: Computational Methods
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 Use the Transpose
operator (‘) to make
a COLUMN Vector
 
>>p = [3,7,9]'
p =
p  7
3
9
7
9
>>g =
[3;7;9]
 Create
Col-Vector g =
3
Directly with
7
SEMICOLONS 9
Bruce Mayer, PE
BMayer@ChabotCollege.edu • ENGR-25_Arrays-1.ppt
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“Appending” Vectors
 Can create a NEW vector by ''appending''
one vector to another
 e.g.; to create the row vector u whose first
three columns contain the values of
r = [2,4,20] and whose 4th, 5th, & 6th
columns contain the values of w = [9,-6,3]
• Typing u = [r,w] yields vector
u = [2,4,20,9,-6,3]
>> r = [2,4,20]; w = [9,-6,3];
>> u = [r,w]
u =
2
4
20
9
-6
Engineering/Math/Physics 25: Computational Methods
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Bruce Mayer, PE
BMayer@ChabotCollege.edu • ENGR-25_Arrays-1.ppt
Colon (:) Operator
 The colon operator (:) easily generates a large
vector of regularly spaced elements.
 Typing >>x = [m:q:n]
• creates a vector x of values with a spacing q. The
first value is m. The last value is n IF m - n is an
integer multiple of q. IF NOT, the last value is
LESS than n.
>> p = [0:2:8]
p =
0
2
4
6
8
>> r = [0:2:7]
r =
0
2
4
6
Engineering/Math/Physics 25: Computational Methods
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Bruce Mayer, PE
BMayer@ChabotCollege.edu • ENGR-25_Arrays-1.ppt
Colon (:) Operator cont.
 To create a row vector z consisting of the
values from 5 to 8 in steps of 0.1,
type z = [5:0.1:8].
 If the increment q is OMITTED, it is taken as
the DEFAULT Value of +1.
>> s =[-11:-6]
s =
-11
-10
-9
>> t = [-11:1:-6]
t =
-11
-10
-9
Engineering/Math/Physics 25: Computational Methods
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-8
-7
-6
-8
-7
-6
Bruce Mayer, PE
BMayer@ChabotCollege.edu • ENGR-25_Arrays-1.ppt
linspace Comand
 The linspace command also creates a
linearly spaced row vector, but instead you
specify the number of values rather than the
increment
 The syntax is linspace(x1,x2,n), where
x1 and x2 are the lower and upper limits and
n is the number of points
• If n is omitted, then it Defaults to 100
 Thus EQUIVALENT statements
>> linspace(5,8,31)
Engineering/Math/Physics 25: Computational Methods
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=
>>[5:0.1:8]
Bruce Mayer, PE
BMayer@ChabotCollege.edu • ENGR-25_Arrays-1.ppt
logspace Comand
 The logspace command creates an array of
logarithmically spaced elements
 Its syntax is logspace(a,b,n), where n is
the number of points between 10a and 10b
 For example, x = logspace(-1,1,4)
produces the vector x = [0.1000,
1
1


0.4642, 2.1544, 10.000] x  10 1 ,10 3 ,10 3 ,101 
>> x = logspace(-1,1,4);
>> y = log10(x)
y =
-1.0000
-0.3333
a  0  b  a   n 1 
a 1 b  a   n 1 
x  10
,10
,10


0.3333
a  2  b  a   n 1 
1.0000
a   n 1 b  a   n 1 
,  ,10
 If n is omitted, the no. of pts defaults to 50
Engineering/Math/Physics 25: Computational Methods
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Bruce Mayer, PE
BMayer@ChabotCollege.edu • ENGR-25_Arrays-1.ppt


logspace Example
 Consider this command
>> y =logspace(-1,2,6)
y =
0.1000
0.3981
1.5849
 Calculate the Power
of 10 increment
p  b  a  n  1
 In this case
p  2   1 6  1
 3 5  0 .6
Engineering/Math/Physics 25: Computational Methods
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6.3096
25.1189
100.0000
 for k = 1→6, then
the kth y-value
y k  10  a  p  k 1
 In this Case
k
Power
yk
1
-1
0.1000
2
-0.4
0.3981
3
0.2
1.5849
4
0.8
6.3096
5
1.4
25.1189
6
2
100.0000
Bruce Mayer, PE
BMayer@ChabotCollege.edu • ENGR-25_Arrays-1.ppt
logspace Example (alternative)
 Consider this command
>> y =logspace(-1,2,6)
y =
0.1000
0.3981
1.5849
6.3096
25.1189
100.0000
1.4000
2.0000
 Take base-10 log of the above
>> yLog10 = log10(y)
yLog10 =
-1.0000
-0.4000
0.2000
0.8000
 Calc Spacing Between Adjacent Elements
with diff command
>> yspc = diff(yLog10)
yspc =
0.6000
0.6000
0.6000
0.6000
0.6000
p  b  a  n  1  2   1 6  1  3 5
Engineering/Math/Physics 25: Computational Methods
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Bruce Mayer, PE
BMayer@ChabotCollege.edu • ENGR-25_Arrays-1.ppt
Vector: Magnitude, Length,
and Absolute Value
 Keep in mind the precise meaning of these
terms when using MATLAB
• The length command gives the number of
elements in the vector
• The magnitude of a vector x having elements x1,
x2, …, xn is a scalar, given by
√(x12 + x22 + … + xn2), and is the
same as the vector's geometric length
• The absolute value of a vector x is, in MATLAB, a
vector whose elements are the arithmetic
absolute values of the elements of x
Engineering/Math/Physics 25: Computational Methods
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Bruce Mayer, PE
BMayer@ChabotCollege.edu • ENGR-25_Arrays-1.ppt
Mag, Length, and Abs-Val
 Geometric Length
 Thus the box diagonal
a x y z
2
 By Pythagoras Find
vector a magnitude
w 2  x 2  y 2 and
a w z
2
2
2
so
a2  x2  y2  z 2
Engineering/Math/Physics 25: Computational Methods
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2
2
>> a =[2,-4,5];
>> length(a)
ans =
3
>> % the Magnitude M =
norm(a)
>> Mag_a = norm(a)
Mag_a =
6.7082
>> abs(a)
ans =
2
4
5
Bruce Mayer, PE
BMayer@ChabotCollege.edu • ENGR-25_Arrays-1.ppt
3D Vector Length Example
% Bruce Mayer * ENGR36
% 25Aug09 * Lec03
% file = VectorLength_0908.m
% NOTE: using “norm” command is much easier
%
% Find the length of a Vector AB with
%% tail CoOrds = (0, 0, 0)
%% tip CoOrds = (2, 1, -5)
%
% Do Pythagorus in steps
vAB = [2 1 -5] % define vector
sqs = vAB.*vAB % sq each CoOrd
sum_sqs = sum(sqs) % Add all sqs
LAB = sqrt(sum_sqs) % Take SQRT of the sumof-sqs
Engineering/Math/Physics 25: Computational Methods
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Bruce Mayer, PE
BMayer@ChabotCollege.edu • ENGR-25_Arrays-1.ppt
3D Vector Length Example - Run
vAB =
2
1
-5
sqs =
4
1
25
sum_sqs =
30
% Bruce Mayer * ENGR36
% 25Aug09 * Lec03
% file = VectorLength_0908.m
%
% Find the length of a Vector AB with
%% tail CoOrds = (0, 0, 0)
%% tip CoOrds = (2, 1, -5)
%
% Do Pythagorus in steps
vAB = [2 1 -5] % define vector
sqs = vAB.*vAB % sq each CoOrd
sum_sqs = sum(sqs) % Add all sqs
LAB = sqrt(sum_sqs) % Take SQRT of the
sum-of-sqs
LAB =
5.4772
Engineering/Math/Physics 25: Computational Methods
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Bruce Mayer, PE
BMayer@ChabotCollege.edu • ENGR-25_Arrays-1.ppt
Matrices/Arrays
 A MATRIX has multiple ROWS and
COLUMNS. For example, the matrix M:
 2 4 10
16 3 7 

M
8 4 9


 3 12 15
 M contains
• FOUR rows
• THREE columns
 VECTORS are SPECIAL CASES of matrices
having ONE ROW or ONE COLUMN.
Engineering/Math/Physics 25: Computational Methods
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Bruce Mayer, PE
BMayer@ChabotCollege.edu • ENGR-25_Arrays-1.ppt
Making Matrices
 For a small matrix you can type it row by row,
separating the elements in a given row with
spaces or commas and separating the rows
with semicolons. For example, typing
 The Command
• >>A =
[2,4,10;16,3,7];
 Generates Matrix
 2 4 10
A

16
3
7


 spaces or commas separate elements in
different columns, whereas semicolons
separate elements in different rows.
Engineering/Math/Physics 25: Computational Methods
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Bruce Mayer, PE
BMayer@ChabotCollege.edu • ENGR-25_Arrays-1.ppt
Making Matrices from Vectors
 Given Row Vectors: r = [1,3,5] and
s = [7,9,11]
 Combine these vectors to form vector-t
and Matrix-D
>> r = [1,3,5]; s = [7,9,11];
>> t = [r s]
t =
1
3
5
7
9
11
>> D = [r;s]
 Note the difference
D =
between the results
1
3
5
7
9
11
given by [r s](or
[r,s]) and [r;s]
Engineering/Math/Physics 25: Computational Methods
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Bruce Mayer, PE
BMayer@ChabotCollege.edu • ENGR-25_Arrays-1.ppt
Directly Make Matrix
 Use COMMAS/SPACES combined with
SEMICOLONS to Construct Matrices
>> D = [[1,3,5]; [7 9 11]]
D =
1
3
5
7
9
11
 Note the use of
• BOTH Commas and Spaces as
COLUMN-Separators
• The SEMICOLON as the ROW-Separator
Engineering/Math/Physics 25: Computational Methods
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Bruce Mayer, PE
BMayer@ChabotCollege.edu • ENGR-25_Arrays-1.ppt
>> r1 = [1:5]
Make Matrix Example
r1 =
1
2
3
4
5
8
9
10
>> r3 = [11:15]
r3 =
11
12
13
14
15
>> r4 = [16:20]
r4 =
16
17
18
19
20
21
22
23
24
>> A = [r1;r2;r3;r4;r5]
25
>> r2 = [6:10]
r2 =
6
7
>> r5 = [21:25]
r5 =
A =
1
6
11
16
21
2
7
12
17
22
3
8
13
18
23
4
9
14
19
24
Engineering/Math/Physics 25: Computational Methods
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5
10
15
20
25
Bruce Mayer, PE
BMayer@ChabotCollege.edu • ENGR-25_Arrays-1.ppt
Array Addressing
 The COLON operator SELECTS individual
elements, rows, columns, or ''subarrays'' of
arrays. Some examples:
• v(:) represents all the row or column elements of
the vector v
• v(2:5) represents the 2nd thru 5th elements; that
is v(2), v(3), v(4), v(5).
• A(:,3) denotes all the row-elements in the third
column of the matrix A.
• A(:,2:5) denotes all the row-elements in the 2nd
thru 5th columns of A.
• A(2:3,1:3) denotes all elements in the 2nd & 3rd
rows that are also in the 1st thru 3rd columns.
Engineering/Math/Physics 25: Computational Methods
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Bruce Mayer, PE
BMayer@ChabotCollege.edu • ENGR-25_Arrays-1.ppt
Array Extraction
 You can use array indices to extract a smaller
array from another array. For example, if you
first create the array B
 2 4 10 13 
16 3 7 18 

B
 8 4 9 25


 3 12 15 17 
 to Produce
SubMatrix C type
C = B(2:3,1:3)
Rows 2&3
Cols 1-3
16 3 7 
C

 8 4 9
Engineering/Math/Physics 25: Computational Methods
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Bruce Mayer, PE
BMayer@ChabotCollege.edu • ENGR-25_Arrays-1.ppt
Array Extraction Example
A =
1
6
11
16
21
2
7
12
17
22
3
8
13
18
23
4
9
14
19
24
>> C = A(3:5,2:5)
C =
12
13
14
17
18
19
22
23
24
15
20
25
Engineering/Math/Physics 25: Computational Methods
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5
10
15
20
25
Bruce Mayer, PE
BMayer@ChabotCollege.edu • ENGR-25_Arrays-1.ppt
Matrix Commands
Command
Description
Computes the arrays u and v, containing the
row and column indices of the NONzero
[u,v,w] =
elements of the matrix A, and the array w,
find(A)
containing the values of the nonzero
elements. The array w may be omitted.
Computes either the number of elements of
length(A) A if A is a vector or the largest value of m or
n if A is an m × n matrix.
Returns a row vector [m n] containing the
size(A)
sizes of the m x n array A.
Engineering/Math/Physics 25: Computational Methods
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Bruce Mayer, PE
BMayer@ChabotCollege.edu • ENGR-25_Arrays-1.ppt
Matrix Commands cont
Command
Description
* Returns the algebraically largest element
in A if A is a VECTOR.
* Returns a row vector containing the largest
max(A) elements in each Column if A is a MATRIX.
* If any of the elements are COMPLEX,
max(A) returns the elements that have the
largest magnitudes.
Similar to max(A) but stores the maximum
[x,k] =
values in the row vector x and their indices
max(A)
in the row vector k.
Engineering/Math/Physics 25: Computational Methods
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Bruce Mayer, PE
BMayer@ChabotCollege.edu • ENGR-25_Arrays-1.ppt
Matrix Commands cont
Command
Description
min(A) and
[x,k] = Like max but returns minimum values.
min(A)
Sorts each column of the array A in
sort(A) ascending order and returns an array the
same size as A.
Sums the elements in each column of the
sum(A) array A and returns a row vector containing
the sums.
Engineering/Math/Physics 25: Computational Methods
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Bruce Mayer, PE
BMayer@ChabotCollege.edu • ENGR-25_Arrays-1.ppt
Examples: Matrix Commands
 Given Matrix A
 6 2
A   10 5
 3 7 
>> A = [[6,2];
[-10,5]; [3,7]];
>> size(A)
ans =
3
2
Engineering/Math/Physics 25: Computational Methods
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>> length(A)
ans =
3
>> max(A)
ans =
6
7
>> min(A)
ans =
-10
2
Bruce Mayer, PE
BMayer@ChabotCollege.edu • ENGR-25_Arrays-1.ppt
WorkSpace Browser
 Allows DIRECT access to Variables for
editing & changing
Engineering/Math/Physics 25: Computational Methods
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Bruce Mayer, PE
BMayer@ChabotCollege.edu • ENGR-25_Arrays-1.ppt
Array Editor
 Permits changing of a single Array Value
without retyping the entire specification
Engineering/Math/Physics 25: Computational Methods
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Bruce Mayer, PE
BMayer@ChabotCollege.edu • ENGR-25_Arrays-1.ppt
MultiDimensional (3) Arrays
 3D (or more-D) Arrays Consist of twodimensional arrays that ar “layered” to
produce a third dimension.
• Each “layer” is called a page.
3D
pg1
Command
pg2
pg3
pg4
Description
Creates a new array by
cat(n,A,B,C, ...) concatenating the arrays A,B,C,
and so on along the dimension n.
Engineering/Math/Physics 25: Computational Methods
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Bruce Mayer, PE
BMayer@ChabotCollege.edu • ENGR-25_Arrays-1.ppt
Vectors Digression
 VECTOR  Parameter Possessing Magnitude
And Direction, Which Add According
To The Parallelogram Law
• Examples: Displacements,
Velocities, Accelerations
 SCALAR  Parameter Possessing
Magnitude But Not Direction
• Examples: Mass, Volume, Temperature
 Vector Classifications
• FIXED or BOUND Vectors Have Well Defined
Points Of Application That CanNOT Be Changed
Without Affecting An Analysis
Engineering/Math/Physics 25: Computational Methods
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Bruce Mayer, PE
BMayer@ChabotCollege.edu • ENGR-25_Arrays-1.ppt
Vectors cont.
• FREE Vectors May Be Moved In Space Without
Changing Their Effect On An Analysis
• SLIDING Vectors May Be Applied Anywhere Along
Their Line Of Action Without Affecting the Analysis
• EQUAL Vectors Have The Same Magnitude
and Direction
• NEGATIVE Vector Of a Given Vector Has The
Same Magnitude but The Opposite Direction
Equal Vectors
Negative Vectors
Engineering/Math/Physics 25: Computational Methods
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Bruce Mayer, PE
BMayer@ChabotCollege.edu • ENGR-25_Arrays-1.ppt
Vector Addition
 Parallelogram Rule
For Vector Addition
 Examine Top & Bottom of
The Parallelogram
• Triangle Rule For
Vector Addition
• Vector Addition is
Commutative
C
B
C
PQ  QP
B
• Vector Subtraction →
Reverse Direction of
The Subtrahend
P  Q  P   Q
Engineering/Math/Physics 25: Computational Methods
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Bruce Mayer, PE
BMayer@ChabotCollege.edu • ENGR-25_Arrays-1.ppt
Vector Addition cont.
 Addition Of Three Or More
Vectors Through Repeated
Application Of The Triangle Rule
 The Polygon Rule For
The Addition Of Three
Or More Vectors
• Vector Addition Is Associative
P  Q  S  P  Q  S  P  Q  S
 Multiplication by a Scalar
• Scales the Vector LENGTH
Engineering/Math/Physics 25: Computational Methods
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Bruce Mayer, PE
BMayer@ChabotCollege.edu • ENGR-25_Arrays-1.ppt
Vector Notation
 In Print and Handwriting We Must
Distinguish Between
• VECTORS
• SCALARS
 These are Equivalent Vector Notations
PPPP
• Boldface Preferred for Math Processors
• Over Arrow/Bar Used for Handwriting
• Underline Preferred for Word Processor
Engineering/Math/Physics 25: Computational Methods
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Bruce Mayer, PE
BMayer@ChabotCollege.edu • ENGR-25_Arrays-1.ppt
Dot Product of 2 Vectors
 The SCALAR PRODUCT or
DOT PRODUCT Between Two
Vectors P and Q Is Defined As
 
P  Q  PQ cos
scalar result 
 Scalar-Product Math Properties
• ARE Commutative
• ARE Distributive
• Are NOT Associative
   
PQ  Q P

 
   
P  Q1  Q2   P  Q1  P  Q2
  
P  Q  S  undefined
– Undefined as (P•S) is NO LONGER a Vector
Engineering/Math/Physics 25: Computational Methods
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Bruce Mayer, PE
BMayer@ChabotCollege.edu • ENGR-25_Arrays-1.ppt
Scalar Product – Cartesian Comps
 Scalar Products With Cartesian Components


 




P  Q  Px i  Py j  Pz k  Qx i  Q y j  Qz k 
 
 
 
>> p = [13,-17,-7];
i i 1 j  j 1 k k 1
 
 
 
q = [-16,5,23];
i  j  0 j  k  0 k  i  0 >> pq = dot(p,q)
 
pq =
P  Q  Px Qx  Py Q y  Pz Qz
-454
 
>> qp = dot(q,p)
P  P  Px2  Py2  Pz2  P 2
 MATLAB Makes this
Easy with the
dot(p,q)
Command
Engineering/Math/Physics 25: Computational Methods
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qp =
-454
>> r = [1 3 5 7];
t = [8,6,4,2];
>> rt = dot(r,t)
rt =
60
Bruce Mayer, PE
BMayer@ChabotCollege.edu • ENGR-25_Arrays-1.ppt
Cross Product
 TWISTING Power of a Force
 MOMENTof the Force
• Quantify Using VECTOR
PRODUCT or CROSS PRODUCT
 Vector Product Of Two Vectors
P and Q Is Defined as The
Vector V Which Satisfies:
• Line of Action of V Is Perpendicular To the
Plane Containing P and Q.
• |V| =|P|•|Q|•sinθ
• Rt Hand Rule Determines Line of Action for V
Engineering/Math/Physics 25: Computational Methods
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Bruce Mayer, PE
BMayer@ChabotCollege.edu • ENGR-25_Arrays-1.ppt
Vector Product Math Properties
 Recall Vector ADDITION Behaved As
Algebraic Addition
– BOTH Commutative and Associative.
 The Vector PRODUCT Math-Properties do
NOT Match Algebra. Vector Products
• Are NOT Commutative
• Are NOT Associative
• ARE Distributive
P Q
Q  P  P  Q   P  Q
NONcommuta tive
P  Q S  P  Q  S 
NONassociative
P  Q1  Q 2   P  Q1  P  Q 2 Distributi ve
Engineering/Math/Physics 25: Computational Methods
39
Q P
Bruce Mayer, PE
BMayer@ChabotCollege.edu • ENGR-25_Arrays-1.ppt
Vector Prod: Rectangular Comps
 Vector Products Of Cartesian Unit Vectors
 
i i  0
  
i j k
 

i k   j
   
 
j  i  k k  i  j
 
 

j j 0
k  j  i
 
  
j k  i
k k  0
 Vector Products In Terms Of
Rectangular Coordinates










V  P  Q  Px i  Py j  Pz k  Qx i  Qy j  Qz k



 Py Qz  Pz Qy i  Pz Qx  PxQz  j  PxQy  Py Qx k



i
j
k
• Summarize Using
P  Q  Px Py Pz
Determinate
Notation
Qx Q y Qz
Engineering/Math/Physics 25: Computational Methods
40
Bruce Mayer, PE
BMayer@ChabotCollege.edu • ENGR-25_Arrays-1.ppt
cross command
 3D Vector (Cross)
product Evaluation
is Painful
• c.f. last Slide
 MATLAB makes it
easy with the
cross(p,q)
command
• The two Vectors
MUST be 3
Dimensional
Engineering/Math/Physics 25: Computational Methods
41
>> p =
>> q =
>> pxq
pxq =
-356
>> qxp
qxp =
356
[13,-17,-7];
[-16,5,23];
= cross(p,q)
-187 -207
= cross(q,p)
187
207
Bruce Mayer, PE
BMayer@ChabotCollege.edu • ENGR-25_Arrays-1.ppt
Array Addition and Subtraction
 Add/Subtract Element-by-Element
p  q  17  9 3   11  4 19  28  5  16
8   15 6 
 6  2  9
AB  





10 3   12 14  2 17 
 Using MATLAB
>> p = [17 -9 3];
>> q = [-11,-4,19];
>> p-q
ans =
28
-5
-16
Engineering/Math/Physics 25: Computational Methods
42
>> A = [6,-2;10,3];
>> B = [9,8;-12,14];
>> A+B
ans =
15
6
-2
17
Bruce Mayer, PE
BMayer@ChabotCollege.edu • ENGR-25_Arrays-1.ppt
Example 2.3-2
x=W
y=N
25’
r
20’
w
r  55i  36 j  25k 
−r
v = w−r
r  55 2  36 2  25 2
v  w  r  w  ( r )
v   20  55i  59  36 j
 20  25k
Engineering/Math/Physics 25: Computational Methods
43
z = Down
Bruce Mayer, PE
BMayer@ChabotCollege.edu • ENGR-25_Arrays-1.ppt
Example 2.3-2 cont
x=W
y=N
r
>> r = [55,36,25];
w
w = [-20,59,15];
v = w-r
>> b = r.*r
b =
3025 1296 625
>> c = sum(b)
z = Down
c =
>> v = w-r
4946
v =
>> dist1 = sqrt(c)
-75
23
-10
dist1 =
>> dist2 =
70.3278
sqrt(sum(v.*v))
dist2 =
79.0822
Engineering/Math/Physics 25: Computational Methods
44
Bruce Mayer, PE
BMayer@ChabotCollege.edu • ENGR-25_Arrays-1.ppt
-r
CAVEAT
 2D arrays are described as
ROWS x COLUMNS
• Elemement of D(2,5)
– 2 → ROW-2
– 5 → COL-5
 ROW Vector →
COMMAS
 COL Vector →
SemiColons
Engineering/Math/Physics 25: Computational Methods
45
>> r = [4,73,17]
r =
4
73
17
>> c = [13;37;99]
c =
13
37
99
Bruce Mayer, PE
BMayer@ChabotCollege.edu • ENGR-25_Arrays-1.ppt
Caveat cont
 Array operations often
CONCATENATE or APPEND
• This Can Occur unexpectedly
• Use the clear command to start from
scratch
concatenate
con·cat·e·nate ( P ) Pronunciation Key (kn-ktn-t, kn-)
tr.v. con·cat·e·nat·ed, con·cat·e·nat·ing, con·cat·e·nates
1. To connect or link in a series or chain.
2. Computer Science. To arrange (strings of characters) into a chained list.
adj. (-nt, -nt)
1. Connected or linked in a series.
Engineering/Math/Physics 25: Computational Methods
46
Bruce Mayer, PE
BMayer@ChabotCollege.edu • ENGR-25_Arrays-1.ppt
All Done for Today
Next Time:
More Array
Operations
Engineering/Math/Physics 25: Computational Methods
47
 Matrix Multiplication
• Explained in Fine
Detail in MATH6
• Some UseFul Links
– http://www.mai.liu.se/~halun/mat
rix/matrix.html
– http://people.hofstra.edu/faculty/
Stefan_Waner/RealWorld/tutoria
lsf1/frames3_2.html
– http://calc101.com/webMathema
tica/matrix-algebra.jspMatrix
Multiplication
Bruce Mayer, PE
BMayer@ChabotCollege.edu • ENGR-25_Arrays-1.ppt
Engr/Math/Physics 25
Appendix-1
Vector Math
Bruce Mayer, PE
Licensed Electrical & Mechanical Engineer
BMayer@ChabotCollege.edu
Engineering/Math/Physics 25: Computational Methods
48
Bruce Mayer, PE
BMayer@ChabotCollege.edu • ENGR-25_Arrays-1.ppt
Rectangular Force Components
 Using Rt-Angle Parallelogram
Resolve F into Perpendicular
Components F  F  F
x
y
 Define Perpendicular UNIT Vectors
Which Are Parallel To The Axes
 Vectors May then Be Expressed As
Products Of The Unit Vectors
With The SCALAR MAGNITUDES
Of The Vector Components
F  Fx  Fy  iFx  jFy
Engineering/Math/Physics 25: Computational Methods
49
Bruce Mayer, PE
BMayer@ChabotCollege.edu • ENGR-25_Arrays-1.ppt
Adding by Components
 Find: Resultant of 3+ Forces
R  PQS
 Plan:
• Resolve Each Force Into Components
• Add LIKE Components To Determine
Resultant
• Use Trig to Fully Describe Resultant
iR x  jRy  iP x  jPy  iQ x  jQy  jS x  jS y
 iP x Q x  S x   jPy  Qy  S y 
Engineering/Math/Physics 25: Computational Methods
50
Bruce Mayer, PE
BMayer@ChabotCollege.edu • ENGR-25_Arrays-1.ppt
Adding by Comp.  cont.
 The Scalar Components Of The Resultant
Are Equal To The Sum Of The
Corresponding Scalar Components Of
The Given Forces
R x  P x  Px Q x
  Fx
R y  Py  Q y  S y
  Fy
 Use The Scalar Components Of The
Resultant to Find the Resultant
Magnitude & Direction By Trig
R R R
2
x
2
y
Engineering/Math/Physics 25: Computational Methods
51
  tan
1
Ry
Rx
Bruce Mayer, PE
BMayer@ChabotCollege.edu • ENGR-25_Arrays-1.ppt
3D Rectangular Components
 The vector F is
contained in the
plane OBAC.
 Resolve F into
horizontal and
vertical
components.
FF
Fy  F cos  y
Fh  F sin  y
Engineering/Math/Physics 25: Computational Methods
52
 Resolve Fh into
rectangular
components
Fx  Fh cos 
 F sin  y cos 
Fz  Fh sin 
 F sin  y sin 
Bruce Mayer, PE
BMayer@ChabotCollege.edu • ENGR-25_Arrays-1.ppt
3D Rectangular Comp. cont.
 With the angles between F and the axes
Fx  F cos  x Fy  F cos  y Fz  F cos  z
F  Fx i  Fy j  Fz k
 F cos  x i  cos  y j  cos  z k 
 Fλ
λ  cos  x i  cos  y j  cos  z k
  is a UNIT VECTOR along the line of
action of F, and cosx, cosy, and cosz,
are the DIRECTION COSINES for F
Engineering/Math/Physics 25: Computational Methods
53
Bruce Mayer, PE
BMayer@ChabotCollege.edu • ENGR-25_Arrays-1.ppt
2-Pt Direction Cosines
 Consider Force, F,
Directed Between Pts
• M(x1,y1,z1)
• N(x2,y2,z2)
 The Vector MN Joins
These Points with
Scalar Components
dx, dy, dz
MN  d  d x i  d y j  d z k
d x  x2  x1
d y  y2  y1
d z  z 2  z1
Engineering/Math/Physics 25: Computational Methods
54
d  d  d x2  d y2  d z2
1
d x i  d y j  d z k 
d
Fd y
Fd x
Fd z
Fx 
Fy 
Fz 
d
d
d
dy
d
d
cos  x  x cos  y 
cos  z  z
d
d
d
F  Fλ 
Bruce Mayer, PE
BMayer@ChabotCollege.edu • ENGR-25_Arrays-1.ppt
Engr/Math/Physics 25
Appendix
Live Demo
Bruce Mayer, PE
Licensed Electrical & Mechanical Engineer
BMayer@ChabotCollege.edu
Engineering/Math/Physics 25: Computational Methods
55
Bruce Mayer, PE
BMayer@ChabotCollege.edu • ENGR-25_Arrays-1.ppt