Vectors and Parametric Equations
PhD. in Mathematics
Norm was 4th at the 2004 USA Weightlifting Nationals! He still trains and competes occasionally, despite his busy schedule.
We are talking about planes in three dimensions. Let's take a look at a problem. The graph below, this graph, shows a rectangular box. Find an equation for the plane that includes, first the bottom of the box.
The bottom of the box looks like it's resting on the x axis and the y axis over here, and it passes through the point 2,0,0, 0,5,0. So what’s common to all points on the bottom of the box, is this z coordinate of 0. So that’s going to be z equals 0 in the equation for the bottom of the box. It’s the xy plane. What about the top? Well assuming that this is a rectangular box, then all four of these points will have the same z coordinate, which is 3. So that’s going to be z equals 3.
How about the right side of the box? If it’s a rectangular box, then all four of these points will have the same y coordinate. They’ll be exactly the same distance away from the xz plane as each other. So they’re all going to have the coordinate 5, and that means that this is going to be y equals 5.
And the front of the box, the front of the box facing us is going to be parallel to the yz plane only it's 2 units closer. So this is going to be the plane x equals 2. Remember that equations tell us the identifying characteristic of the plane. And so when planes are parallel to one of the coordinate planes, they are always going to have nice simple equation like this.
Remember z equals 0 and z equals 3, this is a coordinate plane. z equals 3 is parallel to the xy plane. This one is parallel to the xz plane, the y equals 0, and this one is parallel to the yz plane, x equals 0.