Geoscience Reference
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Figure 2.13. The rectangular coordinate system used in the
addition of rotation vectors. The x-y plane is equatorial with
the x axis passing through 0 Greenwich and the z axis
through the North Pole. Notation and sign conventions are
given in Table 2.3.
z
N
ω
C B
y
λ
CB
φ
CB
x
S
when the three rotation vectors are expressed in their x , y and z components. The
magnitude of the resultant rotation vector, C ω A ,is
x CA + y CA + z CA
ω A =
(2.27)
C
and the pole position is given by
λ CA = sin 1 z CA
C
(2.28)
ω A
and
φ CA = tan 1 y CA
x CA
(2.29)
φ CA has an ambiguity of 180
(e.g., tan 30
Note that this expression for
=
tan 210 =
0.5774, tan 110 =
tan 290 =−
2.747). This is resolved by adding
or subtracting 180
so that
x CA > 0when 90 CA < + 90
(2.30)
x CA < 0when | φ CA | > 90
(2.31)
The problems at the end of this chapter enable the reader to use these methods
to determine motions along real and imagined plate boundaries.
Example: addition of relative rotation vectors
Given the instantaneous rotation vectors in Table 2.1 for the Nazca plate relative to
the Pacific plate and the Pacific plate relative to the Antarctic plate, calculate the
instantaneous rotation vector for the Nazca plate relative to the Antarctic plate.
Rotation
vector
Latitude
of pole
Longitude
of pole
Angular velocity
(10 −7
deg yr −1 )
Plate
55.6 N
90.1 W
Nazca-Pacific
ω N
13.6
P
64.3 S
96.0 E
Pacific-Antarctica
ω P
8.7
A
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