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3-3D Coordinate Systems

3-3D Coordinate Systems - ThreeD Coordinate Systems John E...

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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, 0 , 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 > 0 x y z 4 3 2 -1 Another Viewpoint: x, z > 0 , y < 0
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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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