mthsc206-fall-2010-notes-13.4

mthsc206-fall-2010-notes-13.4 - 13.4 The Cross Product...

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13.4 The Cross Product Algebraic Description Defnition 13.4.1 An m × n matrix is a rectangular array with m rows and n columns of objects. A generic m × n matrix A has the form A = a 11 a 12 a 13 ··· a 1 n a 21 a 22 a 23 a 2 n a 31 a 32 a 33 a 3 n . . . . . . . . . . . . . . . a m 1 a m 2 a m 3 a mn . Defnition 13.4.2 The determinant of a 2 × 2 matrix A = ± a 11 a 12 a 21 a 22 ² , denoted either det( A ) or | A | , is | A | = a 11 a 22 - a 12 a 21 . The determinant of a 3 × 3 matrix A = a 11 a 12 a 13 a 21 a 22 a 23 a 31 a 32 a 33 is given by | A | = a 11 ³ ³ ³ ³ a 22 a 23 a 32 a 33 ³ ³ ³ ³ - a 12 ³ ³ ³ ³ a 21 a 23 a 31 a 33 ³ ³ ³ ³ + a 13 ³ ³ ³ ³ a 21 a 22 a 31 a 32 ³ ³ ³ ³ = a 11 ( a 22 a 33 - a 23 a 32 ) - a 12 ( a 21 a 33 - a 23 a 31 )+ a 13 ( a 21 a 32 - a 22 a 31 ) . Example 13.4.1 Find the determinant of the matrix (a) ± 14 - 27 ² and (b) 1 0 4 0 2 6 2 1 0 . Defnition 13.4.3 Let u = <u 1 ,u 2 3 > and v = <v 1 ,v 2 3 > be two vectors in V 3 . The cross product of u with v , denoted u × v , is given by u × v = ³ ³ ³ ³ ³ ³ i j k u 1 u 2 u 3 v 1 v 2 v
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mthsc206-fall-2010-notes-13.4 - 13.4 The Cross Product...

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