Coordinate Transformations: translation
It is often necessary in science/engineering to express vectors in
different coordinate frames. This requires the rotation and translation
matrixes, which relates coordinates, i.e. basis (unit) vectors in one frame
to those in another frame.
eˆz '
Translation of Coordinate systems
Position coordinate
êz
eˆ y' A A T
'
T
eˆx' A
êy T is the translation vector,
which can be represented by
êx a matrix with 3 elements
coordinate transformations :
rotation about the z-axis
eˆz ' blue: (x, y, z) : ‘old’ system
red: (x’, y’, z) : ‘new’ system
ê z
eˆ y '
θ
ê y
θ
êx
eˆx '
Positive rotation about z axis
(negative rotation would mean clockwise)
Relationship between
various quantities eˆx' cos eˆx sin eˆy
involved in rotation
about the z-axis eˆy' sin eˆx cos eˆy
eˆz' eˆz
eˆ ' cos sin 0 eˆx
x
eˆy' sin cos 0 eˆy
0 1 eˆz
eˆz ' 0
A' x cos sin 0 Ax
'
The components of a vector transform
In the same manner as the coordinates A y sin cos 0 Ay
themselves do, under a coord rotation. A' z 0 0 1 Az
Transformation back from the primed to unprimed
coordinate systesm
eˆx cos sin 0
1 eˆ '
eˆ sin x
y cos 0 eˆy'
eˆz 0 0 1
eˆz'
.
R R RR 1
T T
Transformation matrices are orthogonal so that their
inverse is equal to their transpose
Y
Y/
VY
X/
VY/ VX/
X
V Vx eˆ x V y eˆ y
VX
This can also be written as
V Vx / eˆx / Vy / eˆy /
This can be generalized into three dimensions
We could get the components by taking dot products of unit vectors with V
Vx eˆx V
Vy eˆ y V
Vz eˆz V
Similarly we can get Vx / Vy Vz / /
Vx / eˆx / V eˆx / Vx eˆx Vy eˆy Vz eˆz
Vx eˆ x / eˆ x V y eˆ x / eˆ y Vz eˆ x / eˆ z
Vx / Vx eˆx / eˆx Vy eˆx / eˆy Vz eˆx / eˆz
Vy / Vx eˆy / eˆx Vy eˆy / eˆy Vz eˆy / eˆz
Vz / Vx eˆz / eˆx Vy eˆz / eˆy Vz eˆz / eˆz
This can be written in the matrix form
Vx / eˆx / eˆx eˆx / eˆ y eˆx / eˆz Vx
V e
y y x
/ ˆ / e ˆ eˆ y / eˆ y eˆ y / eˆz V y
V / eˆ / eˆx eˆz / eˆ y eˆz / eˆz Vz
z z
Vx / eˆx / eˆx eˆx / eˆ y eˆx / eˆz Vx
V e
y y x
/ ˆ / e ˆ eˆ y / eˆ y eˆ y / eˆz V y
V / eˆ / eˆx eˆz / eˆ y eˆz / eˆz Vz
z z
similarly
Vx eˆx eˆx / eˆx eˆ y / eˆx eˆz / Vx /
V eˆ eˆ eˆ y eˆ y /
eˆ y eˆz / V y /
y y x /
Vz eˆz eˆx / eˆz eˆ y / eˆz eˆz / Vz /
In general, the transformation matrix can be written
in terms of dot product of unit vectors
A' x eˆx' .eˆx eˆx' .eˆy eˆx' .eˆz AX
'
A y eˆy' .eˆx eˆy' .eˆy eˆy' .eˆz AY
A' z eˆ .eˆ eˆz' .eˆy
A
eˆz' .eˆz Z
z' x
In Einstein summation notation
direction cosine of new unit vector
w.r.t. old unit unit vector
A Rij A j
i
'
Directional cosines
where Rij eˆi ' .eˆ j
Remember: Reflection, rotation, parity etc are
transformations and can be represented by matrices.
Reflection as a transformation
the x axis is reversed;
Reflection along y-z plane : the other axes do not change
eˆz
1 0 0
êz
0 1 0 eˆx
0 0 1
eˆ y
êy
êx
For reflection, det R = −1
Note:
x goes to –x; y and z are unchanged by such a reflection
Reflection transformation
Reflection along y-z plane
1 0 0 eˆz '
0 1 0 êz
0 0 1
eˆx'
êy eˆ y'
êx
Parity Transformation
A parity transformation is the simultaneous flip in the sign of all spatial
coordinates (inversion of co-ordinate system):
A 3×3 matrix representation of P would have determinant equal to -1, and
hence cannot reduce to a rotation.
For a rotation
RT R RRT 1
1 0 0 ê z
eˆx'
and det R = 1
0 1 0 eˆ y ' êy
0 0 1
êx
eˆz '
pseudo vectors or axial vectors
A typical vector (such as the position vector) is transformed
to its negative under inversion of its coordinate axes. Such
"proper" vectors are known as polar vectors. A vector-like
object which is invariant under inversion is called a pseudo
vector, also called an axial vector
Cross Product of the two polar vector is a axial vector
C A B
axial vector (pseudo vector) is a vector which does
not transform like a position vector under reflection
pseudo vectors or axial vectors
A typical vector (such as the position vector) is transformed
to its negative under inversion of its coordinate axes. Such
"proper" vectors are known as polar vectors.
A vector-like object which is invariant under inversion is
called a pseudo vector, also called an axial vector
Cross Product of two polar vectors is an axial vector
Both axial vector (pseudo vector) and polar vector
(normal vector such as position vector) transform in the
same manner, as defined earlier, under proper
rotations.
But axial(pseudo) vectors do not transform like polar
Proper and improper rotations
Geometric rotations vs. parity, inversion and reflection
a pseudovector (or axial vector) is a quantity
that transforms like a vector under a proper rotation,
but gains an additional sign flip under an improper rotation
(Improper rotation:
a transformation that can be expressed as
an inversion followed by a proper rotation).
Pseudo vector carries a sense of rotation or ‘handedness’
of the basis
e.g. angular momentum of a particle has different directions
depending on whether it is moving clockwise or anticlockwise
Polar (proper) vector : gets inverted under 3D inversion
Pseudo (axial) vector : does not change under 3D inversion
3D Inversion (parity transformation) : x -x; y-y and z -z
pseudoscalar denotes a physical quantity analogous to a scalar.
Scalars as well as pseudoscalars are invariant under proper
rotations.
pseudoscalars flip their signs under the parity transformation,
while scalars do not flip, under proper and improper rotations.
Examples:
scalar triple products, magnetic flux, volume
Algebra of Pseudo Vectors and
Examples
Dot and cross products: Examples for axial (pseudo) vectors:
Polar x Polar = Axial
Torque r f
Polar x Axial = Polar
Axial x Axial = Axial Angular Momentum Lrp
Axial . Polar = Pseudo-
scalar Magnetic field
B idl B Biot-Savart's law
B A (more about this in PH102)
Important: A axial vector can never be
equated with a polar vector
Examples :
Some ‘real physical quantities’
Angular Momentum Vector r p
Force on a charged particle moving in an electromagnetic field
F q E v B
(Lorentz Force)
Rotation, reflection and other transformations of vectors is important
Cross-product of two polar vectors is a pseudo-vector
(example: r p )
The Lorentz force, like any other force, is a polar vector,
since it includes the cross-product of a polar vector v
with a pseudo-vector B .
Tensors in Physics and Engineering
Tensors are quantities that transform according to certain rules
under a change of coordinates.
Scalars 2nd rank tensors
• Speed Stress, strain
Conductivity
• Temperature 3rd rank tensors
• Pressure Piezoelectricity
• Energy, Power 4th rank tensors
• Mass, Charge Elastic moduli
• Length,
volume Laws of Physics are represented
in an invariant form,
• Time
though observed quantities
Vectors may be observer - dependent
(tensors of rank 1)
Velocity, Acceleration, Force, nth rank tensor has
Momentum, Displacement 3n components
