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7/30/2019 L01 OCS Post
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LECTURE 1 slide2
Coordinates and Position Vector in RCS
example
1 2 3P x y zOP = + +R a a aJJJG
2 2 1Q x y z OQ = +R a a aJJJG
x
y
z
OPJJJG
Pz
PxP
y
( , , )P P PP x y z
0
POP RJJJG
position coordinates: ( , , )P P Px y zposition vector:
P P x P y P zx y z= + +R a a a
P x P
P y PP z P
xyz
= = =
R aR aR a
7/30/2019 L01 OCS Post
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LECTURE 1 slide3
Principal Planes and Principal Lines in RCS
principal planes satisfy:...
x consty constz const
===
principal lines are intersections
of principal planes:., . ( varies)., . ( varies)., . ( varies)
x const y const zy const z const xz const x const y
= == == =
lineli
line 0,
0,ne 0,
0
00
y z
x
y
z
x=
=
==
=
=Where are these lines?
7/30/2019 L01 OCS Post
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LECTURE 1 slide4
Principal Lines and Base Unit Vectors in RCS
base unit vectors always right-hand triplets
x y z
y z x
z x y
=
=
=
a a a
a a a
a a a
cyclic substitutions apply x y z x
base unit vectors are along principal lines pointing in the directionof coordinate increase
7/30/2019 L01 OCS Post
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LECTURE 1 slide5
Coordinates and Position Vector in CCS
0P P P zz = + +R aa a
x
z( , , )
P P PP z
PP
Pz
position coordinates: ( , , )P P Pz
position vector:1
1
0P
P
P z
= = =
R aR aR a
1 10P zz= + +aR a a
PR
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7/30/2019 L01 OCS Post
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LECTURE 1 slide7
Coordinates and Position Vector in SCS
P
Pr
P
( , , )P P PP r
x
y
z
Pr
P
P
0 0P P rr = + +aR aa
position coordinates: ( , , )P P Pr
position vector:00
P r P
P
P
r
= = =
R aR aR a
ra
a
a
PR
7/30/2019 L01 OCS Post
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LECTURE 1 slide8
(half-plane)
Principal Surfaces and Principal Lines in SCS
Principal Surfaces:
...
r const constconst
==
=
Principal Lines:., . ( varies)., . ( varies)., . ( varies)
r const const const const r const r const
= == == =
line
line
line 1, 9
0
line
, any
145 , 1
c
80 ,
onst.
any cons .0
t
r
r
r
=
= =
==
===
D
D
D
Where are these lines?1
7/30/2019 L01 OCS Post
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LECTURE 1 slide9
Principal Lines and Base Unit Vectors in SCS
r
r
r
=
=
=
a a a
a a a
a a a
base unit vectors:
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7/30/2019 L01 OCS Post
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LECTURE 1 slide11
Q1: In cylindrical CS, the surface = const. is a half-plane.
Q2: In spherical CS, the surface = const. is a sphere.
TRUE OR FALSE?
7/30/2019 L01 OCS Post
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LECTURE 1 slide12
cos
sin
x
yz z
=
==
Rectangular Cylindrical
2 2
arctan
x yy
x
z z
= +
=
=
Transformation of Coordinates
7/30/2019 L01 OCS Post
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LECTURE 1 slide13
2 2 2
2 2 2arccos
arctan
r x y z
z
x y zy
x
= + +
=
+ +
=
Rectangular Spherical
sin cos
sin sincos
x r
y rz r
=
==
Transformation of Coordinates 2
sin
r
sin sinr
x
y
z
cos
r
sin
cos
r
rP
7/30/2019 L01 OCS Post
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LECTURE 1 slide14
Cylindrical Spherical
sin
cos
r
z r
=
==
2 2
2 2arccos
r z
z
z
= +=
= +
Transformation of Coordinates 3
sin
r
cos
r
z
r
7/30/2019 L01 OCS Post
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LECTURE 1 slide15
Q1: The point (0,0,0) in a rectangular CS when transformed
into spherical coordinates result in a coordinate r= 1.
Q2: The point (r,0,0), r 0, in spherical coordinates lies on thex axis.
TRUE OR FALSE?
7/30/2019 L01 OCS Post
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LECTURE 1 slide16
unit vectors of the rectangular coordinate system same atany point of space
More on Unit Vectors 1
x y0
zayaxa
za
yaxaza
ya
xa
1P
2P
3P
7/30/2019 L01 OCS Post
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LECTURE 1 slide17
unit vectors a and a of the CCS NOT the same at differentpoints of space: they depend on
More on Unit Vectors 2
x
y
z
za
1P
2P
a
aza
aa
explain
sin
1
7/30/2019 L01 OCS Post
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LECTURE 1 slide18
unit vectors of the SCS NOT the same at different points ofspace: they depend on and
More on Unit Vectors 3
x
y
z
ra
ra
a
aa
a1P
2P
7/30/2019 L01 OCS Post
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LECTURE 1 slide19
Vectors in RCS
vector components are projections onto the unit vectors of therespective CS at the given position
vector components in general depend on position
( , , ) ( , , ) ( , , )x x y y z zx y z A x y z A x y z= + +A a a a
Aza
y yA a
0x
x
yy
AA+
a
a
xa
ya z za
x xA a
7/30/2019 L01 OCS Post
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LECTURE 1 slide20
Vectors in CCS
a
x
zA
za
a
z
( , , ) ( , , ) ( , , )z zz A z A z = + +A a a a
do not forget that the orientation ofa and a depends on the
position as well
7/30/2019 L01 OCS Post
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LECTURE 1 slide21
Vectors in SCS
( , , ) ( , , ) ( , , )r rr A r A r = + +A a a a
do not forget that the orientation ofar, a, and a depends on the
position as well
x
y
z
A
ra
aa
r
7/30/2019 L01 OCS Post
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LECTURE 1 slide22
cos sin
sin cosx
y
z z
A A A
A A A
A A
= = +
=cos sin
sin cosx y
x y
z z
A A A
A A A
A A
= += +=
vector components depend on the angular position
RCS CCS
Vector Transformations 1
sin
1
/ , ,x x y y z z zA A = + + A a a a a a a
7/30/2019 L01 OCS Post
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LECTURE 1 slide23
RCS SCS
sin cos sin sin cos
cos cos cos sin sin
sin cos
r x y z
x y z
x y
A A A A
A A A A
A A A
= + += +
= +
sin cos cos cos sin
sin sin cos sin cos
cos sin
x r
y r
z r
A A A A
A A A A
A A A
= + = + +
=
vector components depend on the angular coordinates
Vector Transformations 2
7/30/2019 L01 OCS Post
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LECTURE 1 slide24
CCS SCS
Vector Transformations 3
Let us derivesin cos cos cos sin cos sinx r = + = a a a a a a
sin cosr = +a a a
(same result if the ay equation is used)
cos sinz r z = =a a a a
sin cos sin cos cos sincos sin
r r z
z
z r
= + = += = = =
a a a a a aa a a a aa a a a a
sin 0 coscos 0 sin
0 1 0
z
r
a a aaaa
7/30/2019 L01 OCS Post
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LECTURE 1 slide25
CCS SCS
Vector Transformations 4
sin 0 coscos 0 sin
0 1 0
z
r
a a aaa
a
sin cos
cos sin
r
z r
A A A
A AA A A
= +
=
=
sin cos
cos sin
r z
z
A A A
A A AA A
= +
= =
7/30/2019 L01 OCS Post
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LECTURE 1 slide26
A vectorA = 5a exists at the point P( = 1, = 90,
z= 0).
(a) Give the coordinates ofPin RCS.
x = y = z=
(b) What are the components ofA in RCS?
Ax
= Ay
= Az
=
7/30/2019 L01 OCS Post
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LECTURE 1 slide27
Summary
we will be using 3 orthogonal coordinate systems: RCS, CCS, SCS
your can easily transform position coordinates and vector
components formula sheet will be provided in exams papers
the rules of vector multiplication are the same in all orthogonal CSs
the choice of the CS depends on the symmetry of the problem