The Vector or Cross Product of Two Vectors

Thus, the time average of the dot or cross product of two time-harmonic complex quantities is equal to half of the real part of the respective product of one quantity and the complex conjugate of the other. In this regard, the time-averaged Poynting vector is given by... 

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... 

The vector or cross product of two unit vectors is defined by ... 

The vector or cross product of two unit vectors was defined by (C.94), hence in cylindrical coordinates the following relations are valid ... 

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 ... 

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... 

The inner and outer products of two vectors produce a scalar and a matrix, respectively. Two 3-vectors may also be multiplied together to generate a new 3-vector, a procedure called a vector or cross product, with the result given by eq. (16.15). 

We now introduce another kind of a product between two vectors, called the vector product or cross product, and denoted by A x B. If... 

The cross product or vector product of two vectors, denoted A x B or AAB, is a vector that is perpendicular to both A and B and has magnitude lAIIBI sin(0). A x B is oriented in the direction in which a right-handed screw would advance if turning the screw rotates A onto B. Thus AxB and B x A have the same magnitude but point in opposite directions. In vector notation,... 

The cross product or vector product of two vectors is a vector quantity that is perpendicular to the plane containing the two vectors with magnitude equal to the product of the magnitudes of the two vectors times the sine of the angle between them ... 

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