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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 ββ β Β· (β Γ β ) = a v w 1 β2 w 11 = a22 a23 a32 a33 a11 a12 a13 a21 a22 a23 a31 a32 a33 β a12 a21 a23 a31 a33 + a13 a21 a22 a31 a32 . This gives us a geometric interpretation of a 3 Γ 3 (numerical) determinant: Remark 1.7.2. The 3 Γ 3 determinant a11 a12 a13 a21 a22 a23 a31 a32 a33 β ββ β β is the signed volume V ( [β , β 1 , β 2 ]) of the oriented parallelepiped vw w ββ β [β , β 1 , β 2 ] whose generator is the ο¬rst row vw w β β β β β = a β +a β +a β v 13 k 12 11 Δ± and whose base is the oriented parallelogram with edges represented by the other two rows β β β β β = a β +a β +a β w1 23 k 22 21 Δ± β β β β β = a β +a β +a β. w2 33 k 32 31 Δ± For example, the parallelepiped with base OP RQ, with vertices the origin, β β βββ P (0, 1, 0), Q(β1, 1, 0), and R(β1, 2, 0) and generator β = β β β + 2 k v Δ± (Figure 1.54) has βtopβ face OP β² Rβ² Qβ² , with vertices O(1, β1, 2), P β² (1, 0, 2), Qβ² (0, 0, 2) and Rβ² (0, 1, 2). Its signed volume is given by the 3 Γ 3 β ββ β β ββ determinant whose rows are β , OP and OQ: v β β V ( [OP RQ]) = = (1) 1 β1 2 0 10 β1 1 0 10 10 β (β1)(1) 00 β1 0 + (2)(1) = (1)(0) β (β1)(0) + (2)(0 + 1) 01 β1 1 = 2. ...
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