Vector Analysis Notes - Engineering Mathematics

 Page 1


Fall 2006 
Page 2


Fall 2006 
Scalar Quantity 
(mass, speed, voltage, current and power) 
1- Real number (one variable) 
  
2- Complex number (two variables) 
Vector Algebra 
(velocity, electric field and magnetic field) 
Magnitude & direction 
1- Cartesian (rectangular) 
2- Cylindrical 
3- Spherical 
Specified by one of the following 
coordinates best applied to application: 
Page 101 
Page 3


Fall 2006 
Scalar Quantity 
(mass, speed, voltage, current and power) 
1- Real number (one variable) 
  
2- Complex number (two variables) 
Vector Algebra 
(velocity, electric field and magnetic field) 
Magnitude & direction 
1- Cartesian (rectangular) 
2- Cylindrical 
3- Spherical 
Specified by one of the following 
coordinates best applied to application: 
Page 101 
Conventions 
• Vector quantities denoted as    or v 
• We will use column format vectors: 
 
 
 
 
• Each vector is defined with respect to a set of 
basis vectors (which define a co-ordinate 
system). 
v
?
[ ] [ ] ( )
T
v v v v v v
v
v
v
3 2 1 3 2 1
3
2
1
= ?
?
?
?
?
?
?
?
?
?
?
= v
Page 4


Fall 2006 
Scalar Quantity 
(mass, speed, voltage, current and power) 
1- Real number (one variable) 
  
2- Complex number (two variables) 
Vector Algebra 
(velocity, electric field and magnetic field) 
Magnitude & direction 
1- Cartesian (rectangular) 
2- Cylindrical 
3- Spherical 
Specified by one of the following 
coordinates best applied to application: 
Page 101 
Conventions 
• Vector quantities denoted as    or v 
• We will use column format vectors: 
 
 
 
 
• Each vector is defined with respect to a set of 
basis vectors (which define a co-ordinate 
system). 
v
?
[ ] [ ] ( )
T
v v v v v v
v
v
v
3 2 1 3 2 1
3
2
1
= ?
?
?
?
?
?
?
?
?
?
?
= v
Unit vector 
Magnitude 
+
+
+ +
+ +
= =
+ + = =
+ + =
2 2 2
2 2 2
ˆ ˆ ˆ
ˆ
ˆ ˆ ˆ
z y x
z y x
z y x
z y x
A A A
A z A y A x
A
A
a
A A A A A
A z A y A x A
?
?
?
Page 101-102 
a ˆ
Page 5


Fall 2006 
Scalar Quantity 
(mass, speed, voltage, current and power) 
1- Real number (one variable) 
  
2- Complex number (two variables) 
Vector Algebra 
(velocity, electric field and magnetic field) 
Magnitude & direction 
1- Cartesian (rectangular) 
2- Cylindrical 
3- Spherical 
Specified by one of the following 
coordinates best applied to application: 
Page 101 
Conventions 
• Vector quantities denoted as    or v 
• We will use column format vectors: 
 
 
 
 
• Each vector is defined with respect to a set of 
basis vectors (which define a co-ordinate 
system). 
v
?
[ ] [ ] ( )
T
v v v v v v
v
v
v
3 2 1 3 2 1
3
2
1
= ?
?
?
?
?
?
?
?
?
?
?
= v
Unit vector 
Magnitude 
+
+
+ +
+ +
= =
+ + = =
+ + =
2 2 2
2 2 2
ˆ ˆ ˆ
ˆ
ˆ ˆ ˆ
z y x
z y x
z y x
z y x
A A A
A z A y A x
A
A
a
A A A A A
A z A y A x A
?
?
?
Page 101-102 
a ˆ
Vectors 
• Vectors represent directions 
– points represent positions 
 
• Both are meaningless 
without reference to a 
coordinate system 
– vectors require a set of basis 
vectors 
– points require an origin and a 
vector space 
?
?
?
?
?
?
=
?
?
?
?
?
?
=
y
x
y
x
v
v
v
v
P v
both vectors equal