Cross product of two vectors

The vector product (cross product) of two vectors produces a vector... 

Figure 1.39. Vector (cross) product of two vectors. The orientation of V3 is determined using the right-hand rule thumb of the right hand is aligned with Vj, index finger with V2, then V3 is aligned with the middle finger. Tails of all vectors face the middle of the palm.

The vector product (or cross product) of two vectors a and b, written a x b, is a vector c at right angles to the plane of a and b, and equal in magnitude to the product of the absolute values of the two vectors and the sine of the angle a between them, or... 

Figure 17. A, Definition of the cross product of two vectors, B, Description of the triple scalar...

Common examples of pseudo-vectors that will be relevant later include the angular velocity vector f2, the torque T, the vorticity vector co (or the curl of any true vector), and the cross product of two vectors. The inner scaler product of a vector and a pseudo-tensor or a pseudo-vector and a regular tensor will both produce a pseudo-vector. It will also be useful to extend the notion of a pseudo-vector to scalers that are formed as the product of a vector and a pseudo-vector. The third-order, alternating tensor e is a pseudo-tensor of third order as may be verified by reviewing its definition... 

Figure 11.7 shows a diagram of the process in which the format of the data is assembled and the resulting information is obtained. The pure component data matrix is a two-dimensional structure in which the pure array response patterns in each dimension are resolved. The resolution of the original matrix is based on the fact that this matrix is a cross-product of two vectors (the two arrays). 

The curl is a vector and is somewhat analogous to the vector product (cross product) of two vectors. To remember which vector derivative is which, remember that dot and divergence both begin with the letter d and that cross and curl both begin with the letter c. The symbol curlF is sometimes used for the curl ofF. 

The vector product or cross product of two vectors a and b is again a vector ... 

Therefore, the cross product of two vectors in terms of their components can be determined from... 

A vector is not necessarily referred to a Cartesian coordinate system, but, for example, to a neighbor vector with respect to magnitude and direction. Rules about the addition, subtraction, and multiplication (dot and cross products) of two vectors are a part of vector algebra. 

The differential area to be used for localization is given by using a different mathematical operator, the vector product (or cross product) of two vectors. It has the property to give a third vector whose length is equal to the area defined by the two vectors. If W is the result of the cross product of U and V ... 

The cross product of two vectors is also a vector, given by... 

Recall that the cross-product of two vectors separated by 0 results in a third vector that is aligned in a perpendicular direction as the original vectors. For instance, for b x c, the magnimde of the resultant vector would be Ibllcl sin 6, and would be aligned along the a axis. This is in contrast to the dot-product of two vectors, which results in the scalar projection of one vector onto the other, of magnitude Ibllcl cos 0 for b c. For instance, for a cubic unit cell, the denominator terms in Eq. 20, [a (b x c)], would simplify to ... 

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