Lesson 2
Scalars
•
A scalar is a quantity possessing only a magnitude
–
Examples
• Mass, m
• Length,
• Volume, V
• Speed, v
–
–
There is no direction associated with a scalar
Scalars are designated by light-face characters
Vector Representation
•
•
A vector is a quantity possessing a magnitude and a direction
In this course we use two types of vector representations:
–
–
Geometric
Algebraic
Geometric Vectors
•
•
vector
A geometric vector is represented by a
lower-case, light-face, italic character
with an arrow over-score
line-of-action
s
A typical vector s :
–
Magnitude of s is shown as s
or simply as s
–
Examples:
• Position vector: r
• Velocity vector: v
• Acceleration vector: a
• Force vector: f
f
z
a
r
y
x
Unit Vector
•
A unit vector has a magnitude of “1”
unit
–
Example: Unit vector u
v
u
•
A unit vector along the axis of vector
a may be shown as u(a)
•
Any vector can be described as the
product of its magnitude and a unit
vector defined along its axis: a = a u(a)
u(a)
unit magnitude
a
Scalar Product
•
•
•
The scalar (or dot) product of two
vectors is defined as
a. b = ab cos For any two vectors a and b :
a. b = b. a
For any vector a :
a. a = a 2
a
b
Orthogonal vectors:
•
b
Two vectors a and b are said to be
orthogonal (normal or perpendicular)
if:
90 a. b = 0
a
Vector Product
•
Vector (or cross) product of two
vectors a and b is defined as
c = ab
Vector c is perpendicular to the
plane of a and b
The magnitude of c is
computed as c = ab sin For any two vectors a and b :
a b = b a
a
–
–
•
Parallel Vectors
•
In the diagram vectors a , b and c
are parallel
b
Collinear Vectors
•
e
d
In the diagram vectors d and e are
collinear
Algebraic Vectors
•
A 3D vector a in an orthogonal coordinate frame is denoted in algebraic form as a
column array (vector)
a1 a = a2 a 3
–
–
–
An algebraic row vector is written as a T = {a1 a2 a3 }
An algebraic vector is hand-written as a (instead of a)
Note that in the textbook square brackets, [ ] , are used instead of curly, { },
brackets!
Algebraic Vector Operations
•
The figure shows geometric vector
summation and subtraction
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Algebraic vector summation:
c1 a1 + b1 c2 = a2 + b2 c a + b 3 3 3
•
b
c= a+b
d = ab
Algebraic vector subtraction:
c= ab
d1 a1 b1 d2 = a2 b2 d a b 3 3 3
•
Scalar product: a b = {a1 a2
T
–
•
Note: b Ta = a Tb
Skew-symmetric matrix:
b1 a3 } b2 = a1b1 + a2 b2 + a3b3
b 3
b
a
c = a+b
•
a1 0 a3
For a = a2 we define a = a3
0
a a2 a1
3
Vector product: c = a b c = a b
–
•
a2 a1 0 Note: a b = b a and a a = 0
Useful identities (learn how to use them!):
a b = b a T a T b I
~
a b = b a T a b T
= a b b a
a b + a b T = b a + b a T
These identities can be verified by direct calculation (expanding them into
components)
Arrays
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An array is a column vector of any dimension
–
A 6-array (example):
a( x ) a ( y) a(z ) a= 1 2 3 •
Different ways of representing an array:
q1 q 2
q = = {q1
qn •
q2 qn }
T
An array may contain other arrays such as a, b, ..., d with equal or different
dimensions:
a b r= d Matrices
•
A typical m n matrix:
a11
a
21
A=
am1
–
–
•
•
Number or rows: m
Number of columns: n
Square matrix:
m=n
Transpose
Example:
2 1
BT = a 3 2 a
B=
1 3 •
Symmetric matrix
•
Diagonal matrix
A = AT
d11
0
D=
0
•
a1n a2n amn a12
a22
am 2
0
d22
0
0 0 dnn Identity matrix
1 0 0 0 1 0 I=
0 0 1 Matrix Operations
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Summation and Subtraction
•
Multiplication by a scalar
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Multiplication of two matrices
C=A+B
D=AB
E = A
F = AB
Number of columns of A must be equal to the number of rows of B
•
Inverse of a matrix
A 1A = I
–
–
–
–
Inverse of A is denoted as A 1
Matrix must be (i) square and (ii) non-singular
If the inverse does not exist, the matrix is singular
A square singular matrix has redundant rows or columns
Time Derivatives
•
Scalars
d
x x
dt
•
d d
( x ) x
dt dt
Vectors and Arrays
dtd a1 (t ) d
a = a
dt
d a (t ) dt n •
Matrices
dtd b11 d
B = dt
d
dt bm1 •
b1n = B
d
dt bmn d
dt
Chain rule of differentiation:
d
( b ) = b + b
dt
d T
(a b ) = a T b + a T b
dt
= b T a + a T b
d
( a b ) = a b + a b
dt
= b a + a b
d
D+CD
(C D) = C
dt
Partial Derivatives
•
•
•
Assume that q is an n-array of variables
Assume that is a function of q
Partial derivative of (a single function) with respect to q is a row matrix with n
columns ( 1 n ):
=
q q1
•
q2
qn Assume that is an m-array of differentiable functions of q
•
Partial derivative of with respect to q is defined as an m n matrix:
1
q1
q12
=
q m
q1
1
q2
2
q2
m
q2
qn2 qnm 1
qn
•
•
•
Chain rule of differentiation must be applied to partial derivatives with care!
•
The partial derivative of the vector product of a and b with respect to q is found as:
Assume that a and b are n-arrays functions of q
The partial derivative of the scalar product of a and b with respect to q is found as:
T
a
b
(a b ) = b T
+ aT
q
q
q
a
b
( a b ) = b
+ a
q
q
q
Equations and Expressions
•
The term Equation is referred to a set of identities that needs to be solved
simultaneously to find the unknowns
•
The term Expression is referred to one identity or a set of identities that requires only
simple substitutions in order to find the unknowns
Unit System
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•
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In this course the choice of the unit system is left to the student
In most examples the unit system is not specified
In some problems and in the computer programs where the unit system needs to be
specified, the SI units are used