Let a = [a1 a2 a3]T and b = [b1 b2 b3]T be two (general) 3 × 1 single-column matrices.
(I use the device of describing these column matrices as the transposes of certain row matrices simply because HTML makes row matrices much easier to display.)
A 3 × 1 column matrix p = [p1 p2 p3]T is called a cross product of a and b (and we write p = a × b ) if the requirementis valid for all 3 × 1 column matrices c .
p • c = Det( [ a | b | c ] ) (determinant of the 3 × 3 matrix
with columns a , b , and c )
[Recall that by the dot product p • c of column p with column c = [c1 c2 c3]T is just meant the matrix product of the row pT with the column c , i.e., p • c is the single number p • c = p1 c1 + p2 c2 + p3 c3 .]
This will mean thatNow: apply this formula three times, once for c = [1 0 0]T , once for c = [0 1 0]T , and finally also for c = [0 0 1]T , so as to determine what the three entries that make up the column a × b must actually be.
(a × b) • c = Det( [ a | b | c ] ) . (again, determinant of the 3 × 3
matrix with columns a , b , and c )
Note, 2/27/2006. You should have found expressions consistent with those of Section 3.4, formulas (1) or (4).
First posted: 21 Feb 2006.
Last updated: 27 Feb 2006.
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