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Engineering Calculus Notes 120

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

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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 first item, note first that interchanging the two edges of the base reverses the sign of its oriented area and hence the sign of its oriented volume; if the first row is interchanged with one of the other two, you should check that this also reversed the orientation. Once we have the first item, we can assume in the second item that we are scaling the first row, and and in the second that A and B agree except in their first row(s). The additivity and homogeneity in this case follows from the fact that the oriented volume equals the oriented area of the base dotted with the first row. Finally, the last item follows from noting that zero determinant implies zero volume, which means the “height” measured off a plane containing the base is zero. Rotations So far, the physical quantities we have associated with vectors—forces, 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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