4 I The How,When,and Why of Mathematics Fg.1.1 one unit back on the x-axis and one unit in the positive y-direction.Finally the point (2,-1,3)is plotted in Figure 1.2. Let's go a bit further here.In two-space,what wasx=0?Since y does not appear in that equation,it is unrestricted and can be any real number.That's why x=0 in two-space is the y-axis.What is x=3?It is a line parallel to the y-axis through the point(3,0).So,let's try to generalize this to the situation in three-space.What's the planez=0?Recall that if a variable doesn't appear,then it may assume any value. So this means that z is fixed at 0 while x can take any value,as can y.Thus,the plane =0 is the xy-plane.Similarly,the yz-plane is the planex=0 and the xz-plane is the plane y =0.These three planes are called the coordinate planes.What's the plane z=3?x=2?y=yo?There's plenty to think about here,but let's start by asking what the distance is between two points in three-space. Example 1.2.Given two points (xo,yo,zo)and (x1,y1,1)in three-space,what is the distance between the two points? (2,-1,3) (-1,1,0) 1,0,0) Fig.1.24 1 The How, When, and Why of Mathematics x y z Fig. 1.1 one unit back on the x-axis and one unit in the positive y-direction. Finally the point (2,−1,3) is plotted in Figure 1.2. Let’s go a bit further here. In two-space, what was x = 0? Since y does not appear in that equation, it is unrestricted and can be any real number. That’s why x = 0 in two-space is the y-axis. What is x = 3? It is a line parallel to the y-axis through the point (3,0). So, let’s try to generalize this to the situation in three-space. What’s the plane z = 0? Recall that if a variable doesn’t appear, then it may assume any value. So this means that z is fixed at 0 while x can take any value, as can y. Thus, the plane z = 0 is the xy-plane. Similarly, the yz-plane is the plane x = 0 and the xz-plane is the plane y = 0. These three planes are called the coordinate planes. What’s the plane z = 3? x = 2? y = y0? There’s plenty to think about here, but let’s start by asking what the distance is between two points in three-space. Example 1.2. Given two points (x0,y0,z0) and (x1,y1,z1) in three-space, what is the distance between the two points? (1,0,0) (-1,1,0) (2,-1,3) x y z Fig. 1.2