Although we can describe an angular rotation (radians) in the same terms as we do angular velocity, this descriptor is not a vector. Angular rotations do not add commutatively as all vectors do.
CROSS-PRODUCT OF TWO VECTORS
Consideration of the basic definition indicates that the cross-product of parallel vectors is zero (the sine of zero equals zero). The largest possible cross-product of two vectors occurs if the two are perpendicular to one another (the sine of an angle never exceeds one, the value for one half pi). The sense (sign) of the product is reversed if the order of multiplication is reversed. We can show that the cross-product of three or more vectors is associative. This means that we get the same result for any groupings of the vectors as long as the order of multiplication is unchanged. Maybe this paragraph is worse!
For a cartesian coordinate system it's probably easiest to use a 3 by 3 determinant to remember how to determine a cross-product from the components of the two vectors.
ANGULAR ROTATION AS A CROSS-PRODUCT
APPLICATIONS OF CROSS-PRODUCTS
The Lorenz Law describes the ("sideways") force on a electrically charged particle moving through a magnetic field. It is at the heart of how the cathode ray tubes of our televisions and computers work. The Lorenz Law explains the action of mass spectrometers, the principle devices used to "count" the radioactive and stable isotopes so important in geochronology and environmental studies.
The "curl" of a vector field is a vector calculated from the "x", "y" and "z" components of the field by a process similar to calculation of a cross- product. The curl is very important in studying electric fields, magnetic fields, velocity fields in the ocean. You'll see it a lot in advanced geophysics, physical oceanography, fluid mechanics and other important courses.
LITHOSPHERE PLATE VELOCITIES
Each pair of plates has its own Euler pole. Clearly there is a second equivalent Euler pole at the exact opposite point on the globe. There are no topographic or geological features to distinguish an Euler pole from any other point on earth; the poles are mathematical points, not physical entities.
To the extent that a pole (for a pair of plates) doesn't change with time, transform faults (between these two plates) lie on small circles about the Euler pole. On land, transforms such as the San Andreas, North Anatolian, and Dead Sea Faults are marked by numerous earthquakes, horizontally offset rock bodies and a whole complex of geomorphic features. On the ocean floor the transforms produce "fracture zones". These zones have characteristic topography of linear troughs and ridges parallel to the transform motion. Earthquakes are concentrated on the "active" portions of the fracture zones.
The angular velocity of relative plate motion is the same for any point on either plate. Thus a single angular velocity vector is associated with a pair of plates. The vector runs from the center of the earth to the Euler pole. The magnitude is given in units such as radians per million years.
In contrast to the angular velocity, the linear velocity of one plate with respect to another does vary along the plate boundary. This relative velocity is zero at the Euler poles (like the velocity at the center of a wheel) and is greatest half-way between the Euler poles (like the velocity at the rim of a wheel). The linear velocity is simply the cross-product of the angular velocity vector with the location vector of the point on the boundary. This vector is pointed radially from the center of the earth to the boundary point in question.
You can use the program "Nuvel 1" on the web to find relative velocities of any point on any plate with respect to any other plate. The same mathematics is used to find velocities of plates with respect to hot spots and even for hot spots with respect to other hot spots.
Although we can find a single Euler pole for any before and after locations of a pair of plates, there is no reason to suppose that an "effective" Euler pole describes the actual plate trajectory. To accurately represent the true motion history we need to use a series of (different) "stage poles" for each pair of plates.
FINITE ROTATIONS
ADDITION OF FINITE ROTATIONS
An alternative is to use matrix algebra (Cox and Hart, 1986). We will not add finite rotations in this course unless I change my mind.
REFERENCES AND SUGGESTED READING
Le Pichon, X., Francheteau, J. and Bonnin, J., 1973, "Plate Tectonics", Elsevier, 300 pp.
INTRODUCTION
It is conventional to represent angular velocities by vectors. The length of the vector equals the angular velocity. The direction of the vector is given by the "right hand screw rule". The positive sense corresponds to the direction of advance of an ordinary wood screw as you rotate the screw driver clockwise.
We've learned that the "dot" or "scalar" product of two vectors is just a scalar. In contrast, the "cross" or "vector" product of two vectors is itself a vector. The cross product of two vectors (A and B) is a vector (C) with a magnitude equal to the product of the magnitudes of the two vectors (AB) times the sine of the angle between them. The vector C (that is, the cross product vector) is oriented normal to both the two vectors (A and B). Equivalently we can say that the vector C is perpendicular to the plane containing the two original vectors (A and B). The positive direction is given by the right hand rule applied from the first vector (A) to the second(B). This is probably the worst paragraph of the semester!
We can describe an angular rotation of a line in space as the cross-product of the initial orientation of the line with its final orientation. Put the "x", "y", and "z" components of the initial and final vector into the cross-product determinant. Expanding the determinant gives the three components of the rotation (cross-product). We can then find the degree of rotation, the direction cosines and the positive sense of the rotation. Although we are using the terminology of vectors and cross-products, the rotation we just found is not a vector.
An important use of vector cross-products is in describing and calculating plate velocities ("plate kinematics"). Cross-products are very important in study of rotations, torques, angular momentum and related problems in dynamics.
The instantaneous relative motion of a pair of lithosphere plates can be described as a rotation about an Euler pole. Because plates move very slowly "instantaneous" can mean an average condition over perhaps ten million years. These time-averaged Euler poles are called "stage poles".
Unlike angular velocities and infinitesimal (small) rotations, finite (large) rotations can not be added as vectors even though a single rotation (large or small) can be represented by a vector-like descriptor. In particular, a little experimentation shows us that addition of finite rotations is not commutative. This word means that the result (sum) depends on the order in which the rotations are added. Given that true vectors are commutative, we conclude that rotations are not true vectors even though they have a magnitude and direction.
There are at least two ways to add finite rotations. One uses the Olinde Rodriguez Theorem and spherical trigonometry (Le Pichon, 1973). We will not follow this up in this course.
Cox, A. and Hart, R. B., 1086, "Plate Tectonics: How It Works", Blackwell Scientific Publications, 392 pp.