Let= [aa_{1}a_{2}a_{3}]^{T}and= [bb_{1}b_{2}b_{3}]^{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= [pp_{1}p_{2}p_{3}]^{T}is called across productofanda(and we writeb=pa×) if the requirementbis valid for

p•= Det( [c|a|b] )c(determinant of the 3 × 3 matrix

with columns,a, andb)call3 × 1 column matrices.c

[Recall that by thedot productp•of columncwith columnp= [cc_{1}c_{2}c_{3}]^{T}is just meant the matrix product of the rowp^{T}with the column, i.e.,cp•is the single numbercp•=cp_{1}c_{1}+p_{2}c_{2}+p_{3}c_{3}.]

This will mean that

( a×)b•= Det( [c|a|b] ) .c(again, determinant of the 3 × 3

matrix with columns,a, andb)cNow: apply this formula three times, once for= [1 0 0]c^{T}, once for= [0 1 0]c^{T}, and finally also for= [0 0 1]c^{T}, so as to determine what the three entries that make up the columna×must actually be.b

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.[Back to main wes.math.221 page] Contents © 2006 by Fred E.J. Linton