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walkingcarcass

cross product questions

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a) if the length of the cross product of two normalised vectors is the sine of the angle between them, what is the length of the cross product of two arbitary vectors? b) what is the cross product of N-dimensional vectors? ******** A Problem Worthy of Attack Proves It''s Worth by Fighting Back

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a) The length of the cross product of two arbitrary vectors is |v1| * |v2| * sin(v), where v is the angle between the two vectors v1 and v2.

b) Cross product is defined in three dimensional space only.

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Strictly speaking the cross product is defined in only three dimensions, but you can define equivalent function as long as you have more than one dimension. With one dimension you only have one direction so there is no orthogonal direction. In 2D if your vector is (x,y) then an orthogonal vector is (-y,x). That can be viewed as the determinant of the matrix [[x,y],[i,j]] where i and j are the standard basis vectors, i.e. i=[1,0] and j=[0,1]. Needless to say it has the magnitude of the original vector.

Within four dimensions you need three independant vectors for there to be only two vectors orthogonal to all of them. Those two vectors of course have opposite directions. Similar to two the three dimensions you can use the determinant of the matrix [[a1,a2,a3,a4],[b1,b2,b3,b4],[c1,c2,c3,c4],[i,j,k,l]] to calculate one of those vectors. The magnitude seems to be close to the product of the magnitudes and sines, but isn''t exact. Timkin pointed me to a paper once where the author derived what it actually is, but I don''t remember what it was. It seems like it was something that basically makes you say well, imagine that, but not particularly useful. So this is mainly only useful for finding an orthogonal vector in higher dimensions. It is a bit easier than the alternative.

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quote:
Original post by walkingcarcass
what is the determinant of a matrix?

********


A Problem Worthy of Attack
Proves It''s Worth by Fighting Back


determinant of a 2x2 matrix is ad-bc

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[a b]
[c d]


Only square matrices have determinants.

For larger matrices, the process can get complicated really quickly, and google can explain it better than I can

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