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 tonguetwister 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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 Fall '08
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 Calculus, Parallelograms, Vectors, 2 w, Howard Staunton, vw, 13 k, =a − +a, 33 k

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