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Engineering Calculus Notes 125

Engineering Calculus Notes 125 - the sum of the moments of...

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1.7. APPLICATIONS OF CROSS PRODUCTS 113 (b) The faces of the parallelepiped lie on the planes z = 0 z = 1 z = 2 y z = 2 y 1 z = x z = x + 1 . 6. Determine the orientation and volume of the simplex ABCD whose vertices are A (1 , 1 , 1) B (2 , 0 , 1) C (2 , 2 , 1) D (1 , 1 , 0) 7. The plane x + y + z = 3 is continuously rotated about the line x = t y = t z = t (which is perpendicular to the plane and meets it at the point P 0 (1 , 1 , 1)). If the point P (2 , 2 , 1) has velocity −→ v = −→ ı −→ , what is its angular momentum about the line? Challenge problems: 8. Suppose −→ v 0 , −→ v 1 ,..., −→ v n = −→ v 0 are the vertices of an oriented polygon in the plane, traversed in order around the circumference. Show that
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Unformatted text preview: the sum of the moments of the vectors v i − v i − 1 , i = 1 ,... ,n , about the origin is twice the area of the polygon. ( Hint: Compare Exercise 11 and Exercise 13 in the previous section.) 9. Consider the “prism” E bounded below by the xy-plane ( z = 0), above by the plane z = 1, and on the sides by the three vertical planes x = 0 (the yz-plane), y = 0 (the xz-plane), and x + y = 1 (see Figure 1.56 )....
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