Engineering Applications

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Engineering Applications

• Dr. Darrin Leleux
• Lecture 5 Numbers and Vectors
• Chapter 2

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Introduction

• 2.1 Properties of Real Numbers
• 2.2 MATLAB Computer Numbers
• 2.3 Complex Numbers
• 2.4 Vectors in Two Dimensions and Three
  Dimensions

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Topics in Vector Analysis
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2.1 Properties of Real Numbers

• Real numbers consist of
• Integers (, -3, -2, -1, 0, 1, 2, 3, )
• Rational Numbers (Fractions in the form p/q where
  p and q are integers)
• Non-terminating and repeating
• Irrational Numbers cannot be represented
  exactly as a ratio of two integers, ?, e,
  etc.Infinite decimal expansion.
• Non-terminating and non-repeating

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Axioms for Real Numbers

• Axioms or postulates are properties assumed to be
  true
• Theorems are properties derived from axioms
  and/or other theorems.
• Basic axioms define how real numbers are
  combined with addition or multiplication
• Associative laws
• (x y) z x (y z)
• (x ? y) ? z x ? (y ? z)

Subtraction is not associative
(x - y) - z ? x - (y - z)
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Axioms for Real Numbers

• Commutative laws
• The order of addition or multiplication doesnt
  matter for real numbers
• x y y x
• x ? y y ? x
• This will not hold in general for matrix algebra
• Distributive law
• x ? (y z) x ? y x ? z
• Identity Elements
• x 0 x
• 1 ? x x
• Inverse Elements
• x (-x) 0 the negative
• If x ? 0, x ? (1/x) 1 the reciprocal

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Generalization

• All of the values of real numbers are referred to
  as R
• Absolute Value
• The distance of an real number from the origin is
  the magnitude or absolute value
• x, if x ? 0
• -x, if x lt 0

The distance between real numbers x and y is
defined as x y
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2.2 MATLAB Computer Numbers

• Fixed-Point Notation
• Programmer must remember location of decimal
  place
• Floating-Point Notation
• Decimal point floats with the magnitude of the
  number
• MATLAB always works with 64-bit double precision
  numbers
• IEEE Standard 754
• gtgt isieee
• ??? Error using gt isieee
• ISIEEE is obsolete. MATLAB only runs on IEEE
  machines.
• IA-32 Intel Architecture Software Developers
  Manual Volume 1 Basic Architecture
• Section 4.8 has a good discussion of the
  implementation in the Pentium 4 processor (4.12
  MB)
• http//www.intel.com/design/pentium4/manuals/index
  _new.htmsdm_vol1

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IEEE Double Precision Format
ANSI/IEEE Standard 754-1985
sign
fraction
EXAMPLE m-files\bisect.m
Some machines allow these but not all.
Only Used to represent numbers Smaller than
realmin but Greater than zero.
biased exponent
3ff 5 5555 5555 5555 4/3
1.3333
0 011 1111 1111
0 1 0 1 0101 0101 0101 0101
s biased exponent e 1023
2-1 2-2 2-3 2-4
fraction
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IEEE Standard 754
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Special Values

• Indeterminate results 0/0, ?/?
• Results in NaN
• gt 1.79 x 10308 results in Inf
• lt 4.94 x 10-324 results in 0
• Use of denormalized numbers allows numbers to be
  smaller than the theoretical 2.23 x 10-308
• eps 2.2 x 10-16 2-52
• Distance from 1 to the next largest floating
  point number

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Calculating Limits

• Use eps for calculating limits, e.g.
• x 0
• y sin(x)/x
• Results in divide by zero error
• Use eps to perform calculation
• x 0 eps
• y sin(x)/x 1
• How would you verify this?

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lHôpitals Rule

• If
• then,
• , where

Guillaume De l'Hôpital wrote the first calculus
text in 1692.
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Variable Precision Arithmetic

• Use the vpa command in the Symbolic Math Toolbox,
  e.g.
• vpa(pi, 75) will yield p to 75 digits
• vpa allows the user to obtain results of
• calculations to arbitrary precision.

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MATLAB Commands for Real Numbers

• abs Absolute value
• abs(-9.2) 9.2
• ceil Round toward infinity
• ceil(9.2) 10
• fix Round toward zero
• fix(9.2) 9
• floor Round toward minus infinity
• floor(-9.2) -10
• sign Signum function
• sign(-9.2) -1
• sign(9.2) 1
• sign(0.0) 0
• round Round to nearest integer
• round(2.51) 3

How would you perform rounding without the round
command?
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2.3 Complex Numbers

• Polar Form of Complex Numbers
• Roots of Complex Numbers
• MATLAB Complex Numbers

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Complex Numbers (Cartesian)
Argand diagram
x Re z y Im z
In the x-y plane x and y are real numbers and i
is the imaginary unit, where i2 -1
Complex conjugate of z is
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Complex Numbers (Polar)
Principal value of the argument
r z where r is the magnitude of z also
called the modulus or length of z The polar
angle is called the argument of z, thus
In Polar Coordinates
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Properties of Complex Numbers

• z1x1iy1 and z2x2iy2
• (sum) z1z2(x1x2)i(y1y2)
• (negative) -z2 -x2 - iy2
• (subtraction) z1-z2(x1-x2)i(y1-y2)
• (product) z1z2(x1x2-y1y2)i(x1y2y1x2)
• (division)

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Polar Form of Complex Numbers

• z x iy r (cos? i sin? )
• z1z2 r1r2 cos(?1 ?2) i sin(?1 ?2)
• In complex analysis, a complex exponential
  function is defined as ez and is shown to satisfy
  the relationship
• ei? cos(? ) i sin(? ) Eulers formula
• thus, eip i2 and the polar form of a complex
• number can be written as z rei?

, where ? is the argument of z
3 i4 5ei?, ? 53.1
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Roots of Complex Numbers

• Let a be a nonzero complex number.
• Then if zna,
• z is called an Nth root of a.
• De Moivres Theorem (1667-1754)
• (cos ? i sin ? )n cos n? i sin n?
• which follows that if
• a r (cos ? i sin ? ), then

, where p is 0, 1, , n-1
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Example 2.3
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Complex Numbers Example

• Find the roots of z41
• ze?, where ? 2? p/4 and p 0, 1, 2, 3
• thus, ?00, ?1? /2, ?2 ?, and ?3 3? /2
• Therefore,
• z0 e0 1
• z1 ei? /2 i
• z2 ei? -1
• z3 ei3? /2 -i

• Solve z4 1 0
• roots(1 0 0 0 -1)
• ans
• -1.0000
• 0.0000 1.0000i
• 0.0000 1.0000i
• 1.0000

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Complex Number MATLAB Commands

• zxyi, zxyj Complex number
• zrexp(itheta) Polar form
• abs Magnitude
• angle Angle in radians
• conj Complex conjugate
• imag Complex imaginary part y
• real Complex real part x
• compass Draws complex numbers as arrows on
• polar plot
• feather Draws complex numbers as arrows on
• linear plot

Demo compass
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2.4 Vectors in Two Dimensions and Three Dimensions

• Introduction to Vectors
• Dot Product
• Row and Column Vectors
• Projections
• Cross Product

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Introduction to Vectors
Issac Newton (1643-1727)

• 2-D vectors in physics have magnitude and
  direction, e.g.
• pmv
• F ma
• Number in physics that are scalars include
• Power
• Energy
• Vector quantities include
• Velocity
• Acceleration
• Momentum

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Points and Vectors
R2 is used to denote the set of all 2-D
vectors. R3 is used to denote the set of all
3-D vectors. The exponent defines the
dimension of the vectors.
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Example Application of 3-D Vectors

• In Astrodynamics, a 3-D coordinate frame is used
  to define a position vector for a satellite.
• Another 3-D vector is defined to represent the
  velocity of the satellite.

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Addition and Subtraction of Vectors

• x y x1 y1, x2 y2, (addition)
• x y x1 y1, x2 y2, (subtraction)
• x1, x2 by a scalar r is defined by
• rxrx1, rx2

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Addition and Subtraction of vectors
-y
x (-y)
Another way to do vector subtraction is to form
the negative of the vector to be subtracted and
then use vector addition on that vector
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Length of a Vector

• Length of a vector x measures the distance from
  its initial point and terminal point.
• Also called the magnitude.
• Follows from the Pythagorean Theorem
• Also called the norm for vectors in vector space
  R4 or greater.

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Unit Vectors

• A vector x of length 1 is called a unit vector.
• Also called normalized.

Only direction information is preserved. No
magnitude information
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Dot Product

• Dot Product
• Orthogonal Vectors
• Standard Unit Vectors

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Example 2.5
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Example 2.6
motion times the distance traveled and hence
defines the work done.
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Row and Column Vectors

• Dot product representation
• Also called thescalar or inner product

So in MATLAB you can also use xy for the dot
product in Addition to dot(x,y)
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Projections (Fig 2.5)
Projxy
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Projections (cont.)
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Example 2.7
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Cross Product
scalar part
vector part
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Cross Product of Two vectors

• Valid on vectors in R3

• Use right-hand rule to determine
• Direction of cross product.

• Example 2.8, pg. 68