Engineering Calculus Notes 119

Engineering Calculus Notes 119 - 2 case which we noted in...

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1.7. APPLICATIONS OF CROSS PRODUCTS 107 viewed from D , or equivalently if the dot product −−→ AD · ( −−→ AB × −→ AC ) is positive ( resp . negative). In Exercise 9 , we see that the parallelepiped s O PRQ determined by the three vectors −−→ AB , −→ AC , and −−→ AD can be subdivided into six simplices, all congruent to ABCD , and its orientation agrees with that of the simplex. Thus we have Lemma 1.7.3. The signed volume of the oriented simplex ABCD is −→ V ( ABCD ) = 1 6 −−→ AD · ( −−→ AB × −→ AC ) = 1 6 v v v v v v a 11 a 12 a 13 a 21 a 22 a 23 a 31 a 32 a 33 v v v v v v where −−→ AB = a 21 −→ ı + a 22 −→ + a 23 −→ k −→ AC = a 31 −→ ı + a 32 −→ + a 33 −→ k −−→ AD = a 11 −→ ı + a 12 −→ + a 13 −→ k . We can use this geometric interpretation (which is analogous to Proposition 1.6.1 ) to establish several algebraic properties of 3 × 3 determinants, analogous to those in the 2
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Unformatted text preview: 2 case which we noted in 1.6 : Remark 1.7.4. The 3 3 determinant has the following properties: 1. It is skew-symmetric : Interchanging two rows of a 3 3 determinant reverses its sign (and leaves the absolute value unchanged). 2. It is homogeneous in each row: multiplying a single row by a scalar multiplies the determinant by that scalar. 3. It is additive in each row: Suppose two matrices (say A and B ) agree in two rows (say, the two second rows are the same, and the two third rows are the same). Then the matrix with the same second and third rows, but with Frst row equal to the sum of the Frst rows of A and of B , has determinant det A + det B ....
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This note was uploaded on 10/20/2011 for the course MAC 2311 taught by Professor All during the Fall '08 term at University of Florida.

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