ThreeDtext - ThreeD Coordinate Systems John E. Gilbert,...

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Unformatted text preview: ThreeD Coordinate Systems John E. Gilbert, Heather Van Ligten, and Benni Goetz The graph of a function f ( x, y ) of two variables is a surface in 3-space, and familiar single variable methods can be used once a coordinate system has been introduced into 3-space. The xyz-coordinate system in 3-space is obtained by adding the z-axis perpendicular to the usual xy-coordinate system in the xy-plane. Each point P in 3-space is then determined by a triple ( a, b, c ) as shown to the right. The three coordinate axes intersect at the origin, and each pair of axes determines a coordinate plane . If we drop a perpendicular from P to the the xy-plane, we get a point Q with coordinates ( a, b, 0) called the projection of P on the xy-plane. In the same way there are projections R (0 , b, c ) and S ( a, , c ) of P on the yz-plane and zx-plane respectively. x y z P ( a, b, c ) a b c The figures below show the three coordinate planes from two viewpoints : x y z 4 3 2 One Viewpoint: x, y, z > x y z 4 3 2-1 Another Viewpoint: x, z > , y < Colors help identify the coordinate planes: xy-plane { ( x, y, z ) : z = 0 } xz-plane { ( x, y, z ) : y = 0 } yz-plane { ( x, y, z ) : x = 0 } It will be important to understand planes parallel to the these coordinate planes....
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ThreeDtext - ThreeD Coordinate Systems John E. Gilbert,...

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