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.
So we are still talking about planes in the three dimensional coordinate system. I want to point out that the graph of every equation of this form; ax plus by plus cz equals d, is a plane. So if you are asked to graph something like this, 3x plus y plus 2z is equals to 6. The graph is going to be a plane.
So let me give you an idea of how to graph something like this, really easily. In order to do it, is to plot the intercepts first, the x, y and z intercepts, so let’s do that. Now the x intercept is going to be the point where y and z equals 0. It’s a point on the x axis. And so on the x axis y is 0 and z equals 0. So I substitute 0 for y and z, and I get 3x 0,0 equals 6. So x equals 2. And that means my x intercept is right here. So that’s one point.
Then I’m going to plot my y intercept and I get that by plugging in 0 for x and z. So I get 0 plus y plus 0 equals 6, y equals 6. There it is, 1, 2, 3, 4, 5, 6 right here. So that’s 2 and finally the z intercept, I’m going to plug 0 for x and y and I get 0 plus 0 plus 2z equals 6, so z equals 3. And that’s right here.
And so now, the rest is easy. All I have to do, is draw lines connecting these intercepts. That’s true. I don’t get the entire plane but you never get the whole graph. Even when you are graphing a line and you're just getting a little piece of it.
Here I’m getting a triangular piece of the plane, the triangular piece that is contained in the first octane. And so there it is, a graph of the plane 3x plus y plus 2z, equals 6.