Cartesian Three-Space

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+y Q (3, 5, 2)

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Each axis increment is 1 unit

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P ( 5, 4, 3) y

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Figure 7-3 Two points in Cartesian three-space, along

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with the corresponding ordered triples of the form (x,y,z) On all three axes, each increment represents 1 unit Here, we ve gone back to the point of view shown in Figure 7-1

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In Fig 7-3, the coordinates of point P are ( 5, 4,3), and the coordinates of point Q are (3,5, 2) As the system is portrayed here, we can get to point P from the origin by making the following moves in any order: Go 5 units in the negative x direction (straight to the left) Go 4 units in the negative y direction (straight down) Go 3 units in the positive z direction (straight out of the page) We can get from the origin to point Q by doing the following moves in any order: Go 3 units in the positive x direction (straight to the right) Go 5 units in the positive y direction (straight up) Go 2 units in the negative z direction (straight back behind the page) If we were looking at the coordinate grid from a different viewpoint (that of Fig 7-2, for example), our movements would look different, but the points and their coordinates would be the same

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A note for the picayune An ordered triple represents the coordinates of a point in three-space, not the geometric point itself But we may talk or write as if an ordered triple actually is a point, just as we sometimes think of a certain person when we read a name That s okay, as long as we re aware of the semantical difference between the name and the point

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Are you confused

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Some people have trouble envisioning three-dimensional situations in the mind s eye If you re having problems understanding exactly how the three axes should relate in Cartesian three-space, here s a pool rule for the orientation of the axes Imagine the origin of the system resting on the surface of a swimming pool Suppose that we align the positive x axis so that it runs along the water surface, pointing due east Once we ve done that, the other axes are oriented as follows:

Negative values of x are west of the origin Positive values of y are north of the origin Negative values of y are south of the origin Positive values of z are up in the air Negative values of z are under the water

You can look at the coordinate axes from any point you want, whether on the surface, in the sky, or under the water No matter how your view of the system changes, the actual orientation of the axes with respect to each other always stays the same This relative axis orientation is important If it s not strictly followed, we ll get into trouble when we work with graphs and vectors in Cartesian three-space

Here s a challenge!

Imagine an ordered triple (x,y,z) where all three variables are nonzero real numbers Suppose that you ve plotted a point P in xyz space Because x 0, y 0, and z 0, the point P doesn t lie on any of the axes What will happen to the location of P if you

Multiply x by 1 and leave y and z the same Multiply y by 1 and leave x and z the same Multiply z by 1 and leave x and y the same

Solution

Here s what will take place in each of these three situations You can use Fig 7-2 as a visual aid If you re a computer whiz, maybe you can program your machine to create an animated display for each of these three processes: