vectors, arrays and matrices - physics internal website

9/3/2006 Physics 241 -- New Mexico Tech

Vectors, Arrays and MatricesRichard Sonnenfeld

(with some material from W. Palm)

!! VectorsVectorsPhysics definitionPhysics definitionUse in Use in MatlabMatlabProblemsProblems

!! Concept of an ArrayConcept of an Array!! Matrices in Matrices in MatlabMatlab

Matrix multiplicationMatrix multiplication


Vectors and relative motionVectors and relative motion

!! To calculate relative To calculate relative motion or velocity, the motion or velocity, the important quantity is a important quantity is a vector difference.vector difference.

!! The result becomes The result becomes independent of the independent of the choice of coordinate choice of coordinate originorigin

1 2r r−v v

1rv

2rv

Origin 1Origin 2

1 2( )r r−v v2r ′v

1r ′v

1 2r r′ ′−v v

:30


Vector relationships (physics definition)Vector relationships (physics definition)

VectorsVectors!! Allow ready Allow ready

representation of 3 representation of 3 (or more!) (or more!) components at once.components at once.

!! Equations written in Equations written in vector notation are vector notation are more compactmore compact

zdtdzy

dtdyx

dtdx

dtrd ˆˆˆ ++=v

xxxv

=ˆ rrrr vvv ⋅≡≡

∑=

=

≡⋅3

1

)cos(

iii sr

srsr θvv


Matlab can easily represent vectors as defined in the classical physics sense.

The vector r can be specified by three components: x, y, and z, and can be written as:

r = [x, y, z].

However, MATLAB can use vectors having more than three elements.

r


To create a row vector, separate the elements by commas. For example,

>>p = [3,7,9]p =

3 7 9

You can create a column vector by using the transpose notation (').Or by separating the elements by semicolons. For example,

>>p = [3,7,9]'p =

379

>>g = [3;7;9]g =

379


You can create vectors by ''appending'' one vector to another.

For example, if

>> r = [2,4,20] and >> w = [9,-6,3], Then>> u = [r,w]. The result is the vector u = [2,4,20,9,-6,3].You can also create a subset of a vector.Assume we wanted to use the 2nd, 3rd, and 4th elements of u. How could this be done?


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.


For example>> x = [0:2:8]x =

[0,2,4,6,8]

>> x = [0:2:7]x =

[0,2,4,6]

To create 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, then q=1. y = [-3:2]y = [-3,-2,-1,0,1,2].


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 x2are the lower and upper limits and n is the number of points.

For example, linspace(5,8,31) is equivalent to [5:0.1:8].

If n is omitted, the spacing is 1.

Example:t=linspace(0,5);x=3*t-4.9*t.^2;plot(t,x)


Matlab flexibly and cleverly handles row and column vectors and matrices of different sizes.

Most operations are defined on vectors or matrices of the same size. For example, adding a row vector to acolumn vector causes an error.

>> a=[0:10]; b=a’+7; c=a+b

??? Error using ==> +Matrix dimensions must agree.

Get used to looking for these – they are one of the most common causes of strange behavior.

Potential confusionPotential confusion


“Array” (noun) In programming languages, a name given to sequences of variables that have a relationship to eachother.

Examples:Successive points in a scientific data set. Adjacent values of a mathematical function.

Successive array elements are often allocated physically adjacent locations in computer memory.

In Matlab, arrays are referred to as “Vectors” or “Matrices”.

Programming concept called “Array”Programming concept called “Array”


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 x2are the lower and upper limits and n is the number of points.

For example, linspace(5,8,31) is equivalent to [5:0.1:8].

If n is omitted, the spacing is 1.

Example:t=linspace(0,5);x=3*t-4.9*t.^2;plot(t,x)

Vectors and relative motionVectors and relative motion


A car and train move as shown i) Express vTrain and vCar as

vectors.ii) Calculate their relative

velocity vector as well as, its magnitude and direction.

iii) Assume both car and train are 60 miles from the origin and express the distance between them as a time-dependent vector.

iv) Calculate the magnitude of the distance vs. time

v) Plot it and use the plot to determine the time of closest approach.

mph

Relative motion Relative motion (RS5.1)(RS5.1)


Three types of vector (and matrix) multiplication

Element-by-element>> x = [2 3 5 1]>> x.*x (or x.^2)ans =[4 9 25 1]

Inner (dot) product>> x = [2 3 5 1]

>> x*x’ans =

39

Outer-product (not too useful)>> x = [2 3 5 1]>> x’*xans =

4 6 10 26 9 15 310 15 25 52 3 5 1


Three types of vector (and matrix) multiplication

Inner (dot) product>> x = [2 3 5 1]>> x’ ans = 2 3 5 1

>> x*x’ans =

39

Same as matrix product from math[2 3 5 1] [2

3 = 395 1]


Magnitude, Length, and Absolute Value of a Vector

In physics, length, magnitude and absolute value of a vector all mean the same thing. In Matlab, they do not.

Magnitude of a vector x having elements x1, x2, …, xn is a scalar, given by √(x1

2 + x22 + … + xn

2), and is the same as the physics definition of magnitude.

length command gives the number of elements in the vector.

The absolute value of a vector x is a vector whose elements are the absolute values of the elements of x.


For example:>> x = [2,-4,5]

• its length is 3 (length(x))

• its magnitude is √[22 + (–4)2 + 52] = 6.7082 (sqrt(x’*x))

• its absolute value is [2,4,5] (abs(x)).


Vector Relationships Vector Relationships “The dot“The dot--product trick”product trick”

Given vectors Given vectors AA and and B B which correspond to which correspond to symmetry axes of a crystal:symmetry axes of a crystal:

Calculate:Calculate:

Where theta is angle between A and BWhere theta is angle between A and B

xA ˆ2=v

zyxB ˆ3ˆ3ˆ3 ++=v

θ,, BAvv

A

B


Matrices

A matrix has multiple rows and columns. For example, the matrix

has four rows and three columns.

Vectors are special cases of matrices having one row or one column.

M =2 4 10

16 3 7 8 4 9 3 12 15


Creating Matrices

If the matrix is small 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

>>A = [2,4,10;16,3,7];

creates the following matrix:

2 4 1016 3 7

Remember, spaces or commas separate elements in different columns, whereas semicolons separate elements in different rows.

A =


Creating Matrices from Vectors

Suppose a = [1,3,5] and b = [7,9,11] (row vectors). Note the difference between the results given by [a b] and [a;b] in the following session:

>>c = [a b];c =

1 3 5 7 9 11>> D = [[1,3,5];[7,9,11]]

alternatively>>D = [a;b]D =

1 3 57 9 11


Array Addressing

The colon operator selects individual elements, rows, columns, or ''subarrays'' of arrays.

Examples:! v(:) represents all the elements of the vector v.! v(2:5) represents the second through fifth

elements; that is v(2), v(3), v(4), v(5). ! A(:,3) denotes all the elements in the third column

of the matrix A.! A(:,2:5) denotes all the elements in the second

through fifth columns of A.! A(3,:) denotes all the elements in the third row


You can use array indices to extract a smaller array from another array. For example, if you first create the array B

B =

C =16 3 7

8 4 9

2 4 10 1316 3 7 18 8 4 9 253 12 15 17

then type C = B(2:3,1:3), you can produce the following array:


size(A) Returns a row vector [m n]containing the sizes of them x n array A.

sort(A) Sorts each column of the array A in ascending order and returns an array the same size as A.

sum(A) Sums the elements in each column of the array A andreturns a row vector containing the sums.

max(A) ?

Additional Array Functions


The Workspace Browser


The Array Editor


Element-by-element operations: Table 2.3–1

Symbol

+

-

+

-

.*

./

.\

.^

Examples

[6,3]+2=[8,5]

[8,3]-5=[3,-2]

[6,5]+[4,8]=[10,13]

[6,5]-[4,8]=[2,-3]

[3,5].*[4,8]=[12,40]

[2,5]./[4,8]=[2/4,5/8]

[2,5].\[4,8]=[2\4,5\8]

[3,5].^2=[3^2,5^2]

2.^[3,5]=[2^3,2^5]

[3,5].^[2,4]=[3^2,5^4]

Operation

Scalar-array addition

Scalar-array subtraction

Array addition

Array subtraction

Array multiplication

Array right division

Array left division

Array exponentiation

Form

A + b

A – b

A + B

A – B

A.*B

A./B

A.\B

A.^B


The array operations are performed between the elements in corresponding locations in the arrays. For example, the array multiplication operation A.*B results in a matrix C that has the same size as A and B and has the elements ci j = ai j bi j . For example, if

then C = A.*B gives this result:

A = 11 5–9 4

B = –7 86 2

C = 11(–7) 5(8)–9(6) 4(2)

= –77 40–54 8


Matrix-Matrix Multiplication

In the product of two matrices AB, the number of columns in A must equal the number of rows in B. The row-column multiplications form column vectors, and these column vectors form the matrix result. The product AB has the same number of rows as A and the same number of columns as B. For example,

6 –2 10 3

4 7

9 8–5 12 =

64 24 75 1161 116

= (2.4–4)

(6)(9) + (– 2)(– 5) (6)(8) + (– 2)(12)(10)(9) + (3)(– 5) (10)(8) + (3)(12)

(4)(9) + (7)(– 5) (4)(8) + (7)(12)


Use the operator * to perform matrix multiplication in MATLAB. The following MATLAB session shows how to perform the matrix multiplication shown in (2.4–4).

>>A = [6,-2;10,3;4,7];>>B = [9,8;-5,12];>>A*Bans =

64 2475 1161 116


Matrix multiplication does not have the commutative property; that is, in general, AB ≠ BA. A simple example will demonstrate this fact:

AB = 6 –210 3

9 8–12 14

= 78 2054 122

(2.4–6)

BA = 9 8–12 14

6 –210 3

= 134 668 65 (2.4–7)

whereas

Reversing the order of matrix multiplication is a common and easily made mistake.

vectors, arrays and matrices - physics internal website

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