Engineering Calculus Notes 120

Engineering Calculus Notes 120 - displacements . In e²ect,...

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108 CHAPTER 1. COORDINATES AND VECTORS 4. A 3 × 3 determinant equals zero precisely if its rows are linearly dependent. For the frst item, note frst that interchanging the two edges o± the base reverses the sign o± its oriented area and hence the sign o± its oriented volume; i± the frst row is interchanged with one o± the other two, you should check that this also reversed the orientation. Once we have the frst item, we can assume in the second item that we are scaling the frst row, and and in the second that A and B agree except in their frst row(s). The additivity and homogeneity in this case ±ollows ±rom the ±act that the oriented volume equals the oriented area o± the base dotted with the frst row. Finally, the last item ±ollows ±rom noting that zero determinant implies zero volume, which means the “height” measured o² a plane containing the base is zero. Rotations So ±ar, the physical quantities we have associated with vectors—±orces, velocities—concern
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Unformatted text preview: displacements . In e²ect, we have been talking about the motion o± individual points, or the abstraction o± such motion ±or larger bodies obtained by replacing each body with its center o± mass. However, a complete description o± the motion o± solid bodies also involves rotation . A rotation o± 3-space about the z-axis is most easily described in cylindrical coordinates: a point P with cylindrical coordinates ( r,θ,z ), under a counterclockwise rotation (seen ±rom above the xy-plane) by α radians does not change its r- or z- coordinates, but its θ- coordinate increases by α . Expressing this in rectangular coordinates, we see that the rotation about the z-axis by α radians counterclockwise (when seen ±rom above) moves the point with rectangular coordinates ( x,y,z ), where x = r cos θ y = r sin θ to the point x ( α ) = r cos( θ + α ) y ( α ) = r sin( θ + α ) z ( α ) = z....
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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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