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 requirement

p • c = Det(   [ a | b | c ]   )

(determinant of the 3 × 3 matrix with columns a , b , and c )

is valid for all 3 × 1 column matrices 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 p^T with the column c , i.e., p • c is the single number p • c = p₁ c₁ + p₂ c₂ + p₃ c₃ .]

  This will mean that

(a × b) • c = Det(   [ a | b | c ]   ) .

(again, determinant of the 3 × 3 matrix with columns a , b , and c )

  Now: 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.

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  Note, 2/27/2006. You should have found expressions consistent with those of Section 3.4, formulas (1) or (4).

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First posted: 21 Feb 2006. Last updated: 27 Feb 2006.

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