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VECTOR REPRESENTATION
3 PRIMARY COORDINATE SYSTEMS:
• RECTANGULAR
• CYLINDRICAL
• SPHERICAL
Choice is based on symmetry of problem
Examples:
Sheets - RECTANGULAR
Wires/Cables - CYLINDRICAL
Spheres - SPHERICAL
Orthogonal Coordinate Systems: (coordinates mutually perpendicular)
Spherical Coordinates
Cylindrical Coordinates
Cartesian Coordinates
P (x,y,z)
P (r, Θ, Φ)
P (r, Θ, z)
x
y
zP(x,y,z)
θ
z
rx y
z
P(r, θ, z)
θ
Φ
r
z
yx
P(r, θ, Φ)
Page 108
Rectangular Coordinates
Cartesian CoordinatesP(x,y,z)
Spherical CoordinatesP(r, θ, Φ)
Cylindrical CoordinatesP(r, θ, z)
x
y
zP(x,y,z)
θ
z
rx y
z
P(r, θ, z)
θ
Φ
r
z
yx
P(r, θ, Φ)
Page 109
x
y
z
Z plane
y planex plane
xyz
x1
y1
z1
Ax
Ay
Unit vector properties
0ˆˆˆˆˆˆ
1ˆˆˆˆˆˆ
xzzyyx
zzyyxx
yxz
xzy
zyx
ˆˆˆ
ˆˆˆ
ˆˆˆ
Vector Representation
Unit (Base) vectors
A unit vector aA along A is a vector whose magnitude is unity
A
Aa
zyx AzAyAxA ˆˆˆ
Page 109
x
y
z
Z plane
y planex plane
222zyx AAAAAA
xyz
x1
y1
z1
Ax
Ay
Az
Vector representation
Magnitude of A
Position vector A
),,( 111 zyxA
111 ˆˆˆ zzyyxx
Vector Representation
x
y
z
Ax
Ay
AzA
B
Dot product:
zzyyxx BABABABA
Cross product:
zyx
zyx
BBB
AAA
zyx
BA
ˆˆˆ
Back
Cartesian Coordinates
Page 108
x
z
y
VECTOR REPRESENTATION: UNIT VECTORS
yaxa
zaUnit Vector
Representation for Rectangular
Coordinate System
xaThe Unit Vectors imply :
ya
za
Points in the direction of increasing x
Points in the direction of increasing y
Points in the direction of increasing z
Rectangular Coordinate System
r
f
z
P
x
z
y
VECTOR REPRESENTATION: UNIT VECTORS
Cylindrical Coordinate System
za
a
ra
The Unit Vectors imply :
za
Points in the direction of increasing r
Points in the direction of increasing j
Points in the direction of increasing z
ra
a
BaseVectors
A1
ρ radial distance in x-y plane
Φ azimuth angle measured from the positive x-axis
Z
r0
20
z
Cylindrical Coordinates
ˆˆˆ
,ˆˆˆ
,ˆˆˆ
z
z
z
zAzAAAaA ˆˆˆˆ
Pages 109-112Back
( ρ, Φ, z)
Vector representation
222zAAAAAA
Magnitude of A
Position vector A
Base vector properties
11 ˆˆ zz
Dot product:
zzrr BABABABA
Cross product:
zr
zr
BBB
AAA
zr
BA
ˆˆˆ
B A
Back
Cylindrical Coordinates
Pages 109-111
VECTOR REPRESENTATION: UNIT VECTORS
Spherical Coordinate System
r
f
P
x
z
y
q
a
a
ra
The Unit Vectors imply :
Points in the direction of increasing r
Points in the direction of increasing j
Points in the direction of increasing q
ra
aa
ˆˆˆ,ˆˆˆ,ˆˆˆ RRR
Spherical Coordinates
Pages 113-115Back
(R, θ, Φ)
AAARA Rˆˆˆ
Vector representation
222 AAAAAA R
Magnitude of A
Position vector A
1ˆRR
Base vector properties
Dot product:
BABABABA RR
Cross product:
BBB
AAA
R
BA
R
R
ˆˆˆ
Back
B A
Spherical Coordinates
Pages 113-114
zr aaa ˆˆˆ aaar ˆˆˆ zyx aaa ˆˆˆ
RECTANGULAR Coordinate Systems
CYLINDRICAL Coordinate Systems
SPHERICAL Coordinate Systems
NOTE THE ORDER!
r,f, z r, q ,f
Note: We do not emphasize transformations between coordinate systems
VECTOR REPRESENTATION: UNIT VECTORS
Summary
METRIC COEFFICIENTS
1. Rectangular Coordinates:
When you move a small amount in x-direction, the distance is dx
In a similar fashion, you generate dy and dz
Unit is in “meters”