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We can create three-dimensional graphs by adding a third axis perpendicular to the familiar x and y axes of the Cartesian plane The new axis, usually called the z axis, passes through the xy plane at the origin, giving us Cartesian three-space or Cartesian xyz space
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Cartesian three-space has three real-number lines positioned so they all intersect at their zero points, and so each line is perpendicular to the other two The point where the axes intersect constitutes the origin Each axis portrays a real-number variable
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Axes and variables Figure 7-1 is a perspective drawing of a Cartesian xyz space coordinate system In a true-to-life three-dimensional portrayal, the positive x axis would run to the right, the negative x axis would run to the left, the positive y axis would run upward, the negative y axis would run downward, the positive z axis would project out from the page toward us, and the negative z axis would project behind the page away from us In Cartesian three-space, the axes are all linear, and they re all graduated in increments of the same size For any axis, the change in value is always directly proportional to the physical displacement If we move 3 millimeters along an axis and the value changes by 1 unit, then that s true all along the axis, and it s also true everywhere along both of the other axes If the divisions differ in size between the axes, then we have rectangular three-space, but not true Cartesian three-space Cartesian three-space is often used to graph relations and functions having two independent variables When this is done, x and y are usually the independent variables, and z is the dependent variable, whose value depends on both x and y
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Figure 7-1 A pictorial rendition of Cartesian three-space
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In this view, the x axis increases positively from left to right, the y axis increases positively from the bottom up, and the z axis increases positively from far to near
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Biaxial planes Cartesian three-space contains three flat biaxial (two-axis) planes that intersect along the coordinate axes
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The xy plane contains the axes for the variables x and y The xz plane contains the axes for the variables x and z The yz plane contains the axes for the variables y and z You ll see three rectangles in Fig 7-2, one parallel to each of the three biaxial planes Look closely at how these rectangles are oriented They can help you envision the orientations of the three biaxial planes in space Each of the three biaxial planes is perpendicular to both of the others
Are you astute
Figure 7-2 shows an alternative perspective on Cartesian three-space, in which we re looking up toward the xz plane from somewhere near the negative y axis There s a difference between the apparent positions of the axes in Fig 7-2 as compared with their positions in Fig 7-1, but the orientations of the three axes are the same with respect to each other You should get used to seeing Cartesian three-space from various points of view I ll switch points of view often to keep you thinking!
How It s Assembled
+z Rectangle parallel to xz plane
Rectangle parallel to yz plane
Rectangle parallel to xy plane
Figure 7-2 Cartesian three-space contains the xy, xz, and
yz planes This drawing shows rectangles parallel to each of these three biaxial planes Note the difference in the point of view between this illustration and Figure 7-1
Points and ordered triples Figure 7-3 shows two specific points P and Q, plotted in Cartesian three-space We ve returned to the perspective of Fig 7-1, with the positive z axis coming out of the page toward us A point can always be denoted as an ordered triple in the form (x,y,z), according to the following scheme:
The x coordinate represents the point s projection onto the x axis The y coordinate represents the point s projection onto the y axis The z coordinate represents the point s projection onto the z axis We get the projection of a point onto an axis by drawing a line from that point to the axis, and making sure that the line intersects that axis at a right angle If this notion gives you trouble, you can think of the x, y, and z values for a particular point in the following way: The x coordinate is the point s perpendicular displacement (positive, negative, or zero) from the yz plane The y coordinate is the point s perpendicular displacement (positive, negative, or zero) from the xz plane The z coordinate is the point s perpendicular displacement (positive, negative, or zero) from the xy plane
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