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Engineering Calculus Notes 116

Engineering Calculus Notes 116 - 104 CHAPTER 1 COORDINATES...

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Unformatted text preview: 104 CHAPTER 1. COORDINATES AND VECTORS parallel parallelograms,20 called the faces. If the base parallelogram has → → → sides represented by the vectors − 1 and − 2 and the generator is − w w v →→ → vw w (Figure 1.53) we denote the parallelepiped by [− , − 1 , − 2 ]. The oriented → − v → − w2 → − w 1 Figure 1.53: Parallelepiped area of the base is → → A (B ) = − 1 × − 2 w w so the signed volume is21 → −− − − → →→ → V ( [→, →1 , →2 ]) = − · A (B ) = − · (− 1 × − 2 ) vw w v v w w → (where − represents the third edge, or generator). v If the components of the “edge” vectors are → → − =a − +a − +a − → → v 11 ı 12 13 k → → − =a − +a − +a − → → w ı k 1 21 22 23 → → − =a − +a − +a − → → w2 31 ı 32 33 k then → − ×− = w 1 →2 w →→→ −−− ı k a21 a22 a23 a31 a32 a33 → =− ı 20 a22 a23 a32 a33 → −− a21 a23 a31 a33 → − a21 a22 +k a31 a32 This tongue-twister was unintentional! :-) The last calculation in this equation is sometimes called the triple scalar product →→ → of − , − 1 and − 2 . vw w 21 ...
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