lect10_4

lect10_4 - The Cross Product - (10.4) 1. Determinant of a...

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The Cross Product - (10.4) 1. Determinant of a Matrix A2 ! 2 matrix and 3 ! 3 matrix of real numbers are of the forms a 11 a 12 a 21 a 22 and a 11 a 12 a 13 a 21 a 22 a 23 a 31 a 32 a 33 , respectively. The determinant of a 2 ! 2 matrix of real numbers is defined by a 11 a 12 a 21 a 22 " a 11 a 22 ! a 12 a 21 , and the determinant of a 3 ! 3 matrix of real numbers is defined by a 11 a 12 a 13 a 21 a 22 a 23 a 31 a 32 a 33 " a 11 a 22 a 23 a 32 a 33 ! a 12 a 21 a 23 a 31 a 33 # a 13 a 21 a 22 a 31 a 32 . You will study the determinant of a higher order square matrix in a linear algebra course. Example Evaluate the determinants 1 ! 2 ! 34 , and ! 10 2 5 ! 43 ! 60 ! 8 . 1 ! 2 ! " ! 1 "! 4 " ! ! ! 2 "! ! 3 " " ! 2 ! 5 ! ! ! 8 " ! ! 1 " ! 0 ! 8 ! 0 # 2 5 ! 4 ! " ! ! 1 "! ! 4 "! ! 8 " # 2 ! 0 ! ! ! 4 "! ! 6 "" " ! 80, 2. The Cross Product Let u $ "% u 1 , u 2 , u 3 & , and v $ "% v 1 , v 2 , v 3 & . Then the cross product of u $ and v $ are defined by u $ ! v $ " i $ j $ k $ u 1 u 2 u 3 v 1 v 2 v 3 " u 2 u 3 v 2 v 3 i $ ! u 1 u 3 v 1 v 3 j $ # u 1 u 2 v 1 v 2 k $ . Example Let u ! "% 1, ! 2,3 & and v ! "% 4,1,2 & . Compute w ! " u ! ! v ! and check w ! # u ! and w ! # v ! . w ! " u ! ! v ! " i ! j ! k ! 1 ! 23 412 " ! 12 i ! ! 13 42 j ! # 1 ! 2 41 k ! " ! ! 4 ! 3 " i ! ! ! 2 ! 12 " j ! # ! 1 # 8 " k ! " ! 7 i ! # 10 j ! # 9 k ! 1
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w ! # u ! " ! 7 i ! # 10 j ! # 9 k ! # i ! ! 2 j ! # 3 k ! " ! ! 7 " # ! ! 20 " # 27 " 0 w ! # v ! " ! 7 i ! # 10 j ! # 9 k ! # 4 i ! # j ! # 2 k ! " ! ! 28 " # 10 # 18 " 0 Note that w $ is orthogonal to both u $ and v $ and therefore it is orthogonal to the plane obtained by vectors u $ and v $ . 3. Properties of the Cross Product Let u $ , v $ and w $ be vectors in V 3 and c be a scalar. Then a. u $ !
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lect10_4 - The Cross Product - (10.4) 1. Determinant of a...

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