Engineering Calculus Notes 117

Engineering Calculus Notes 117 - 105 1.7. APPLICATIONS OF...

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Unformatted text preview: 105 1.7. APPLICATIONS OF CROSS PRODUCTS so →→ − · (− × − ) = a v w 1 →2 w 11 = a22 a23 a32 a33 a11 a12 a13 a21 a22 a23 a31 a32 a33 − a12 a21 a23 a31 a33 + a13 a21 a22 a31 a32 . This gives us a geometric interpretation of a 3 × 3 (numerical) determinant: Remark 1.7.2. The 3 × 3 determinant a11 a12 a13 a21 a22 a23 a31 a32 a33 → →→ → − is the signed volume V ( [− , − 1 , − 2 ]) of the oriented parallelepiped vw w →→ → [− , − 1 , − 2 ] whose generator is the first row vw w → → → → − = a − +a − +a − v 13 k 12 11 ı and whose base is the oriented parallelogram with edges represented by the other two rows → → → → − = a − +a − +a − w1 23 k 22 21 ı → → → → − = a − +a − +a −. w2 33 k 32 31 ı For example, the parallelepiped with base OP RQ, with vertices the origin, → − →→→ P (0, 1, 0), Q(−1, 1, 0), and R(−1, 2, 0) and generator − = − − − + 2 k v ı (Figure 1.54) has “top” face OP ′ R′ Q′ , with vertices O(1, −1, 2), P ′ (1, 0, 2), Q′ (0, 0, 2) and R′ (0, 1, 2). Its signed volume is given by the 3 × 3 − −→ − → −→ determinant whose rows are − , OP and OQ: v → − V ( [OP RQ]) = = (1) 1 −1 2 0 10 −1 1 0 10 10 − (−1)(1) 00 −1 0 + (2)(1) = (1)(0) − (−1)(0) + (2)(0 + 1) 01 −1 1 = 2. ...
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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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