Cross Products; Rotating in Three DimensionsDate: 10/26/2001 at 00:00:24 From: RJ Subject: Matrices (Cross-Product) Hi, We are currently studying matrices and vectors, and we just came to the concept of the cross product. Our entire class understands how it works, but we don't understand why or the proof behind it. Could you give us any insight as to why it works? Thanks. Date: 10/26/2001 at 03:37:33 From: Doctor Mitteldorf Subject: Re: Matrices (Cross-Product) Dear RJ, One important thing to know about cross-products is that they're a special accident of 3 dimensions. The cross-product of two vectors is about rotation in the plane defined by those two vectors. Components of the product are not really vector components - they're really directions of a plane surface. The three components of a cross product are really not (x,y,z) but rather (yz,zx,xy). If we lived in a 4-dimensional world, there would be a cross-product object, but it wouldn't be a vector. It would be a six-component thing, called a "tensor," with all possible pairs of the 4 coordinates for components: wx,wy,wz,xy, and xz. (w1) (w2) ( 0 w2x1-x2w1 w2y1-y2w1 w2z1-z2w1) (x1) X (x2) = (w1x2-x1w2 0 x2y1-y2x1 x2z1-z2x1) (y1) (y2) (w1y2-y1w2 x1y2-x2y1 0 y2z1-z2y1) (z1) (z2) (w1z2-z1w2 x1z2-z1x2 y1z2-z1y2 0 ) In other words, How many ways are there to rotate in 3 dimensions? There are 3, but it's only an accident that this number is the same as the number of dimensions. It's really the number of pairs of different dimensions that is the answer, and in 3 dimensions this happens to be the same number. In 4 dimensions, there aren't 4 pairs of dimensions, but 6. (How many such pairs are there in 5 dimensions?) So it's a shorthand and a convenience that we write the cross-product as a vector, rather than as an "anti-symmetric tensor." Can you induce what the word "anti-symmetric" means here? Think of the cross product of vector A and vector B as a rotation in the plane defined by A and B. ...But maybe this isn't what you were looking for. Please write back and tell me what you meant by "how it works" and whether there's a specific proof you don't understand. - Doctor Mitteldorf, The Math Forum http://mathforum.org/dr.math/ |
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