Date: Summer 2013
I know that the magnitude of the cross product of 2 vectors is the magnitude of vector A times the magnitude of vector B times the sin of the angle between the vectors . It is generally expected that you use the right hand rule to find the direction of the cross product. Is there however, any numerical way to find the direction or to find the unit vector that is orthogonal to both vector A and vector B? I have seen and heard many variations of the right hand rule and I can never seem to position my hand right. Having a numerical way would be much easier for people like me who are bad with their hands.
The numerical way to find out the direction of a cross product is
actually to perform the cross-product operation. All that right-hand
rule stuff is intended to simplify the operation by making it more
easily visualized. If it does not help, do not worry about it.
You can very easily visualize at least the line along which the cross
product must lie, just by thinking of vectors A and B as directions on
the surface of Earth: North and East, for instance, or maybe North
and Northwest, or whatever. The cross product will in any case be
vertical - either up or down, depending on the right hand rule.
The standard numerical way to find the cross product AxB of vectors A and B is to make a 3x3 matrix of their cartesian components with the unit vectors on the first line, vector A on the second line, and vector B on the third line, as follows:
i j k
Ax Ay Az
Bx By Bz
The determinant of this matrix is the vector AxB.
There is no shortcut I know of to find the unit vector in this direction; as far as I know you have to find the vector's magnitude and divide.
Richard E. Barrans Jr., Ph.D., M.Ed.
Department of Physics and Astronomy
University of Wyoming
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Update: November 2011