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.
When introduced to the 3D coordinate system we are introduced to the concept of math planes, and vector equations for planes. Math planes are used frequently with vectors, when calculating normal vectors to planes or when finding the angle between two planes. It is simpler to find the equations of math planes that is formed by two axes, or a plane that is parallel to one.
I want to talk about planes, the 3 dimensional coordinates system is loaded with them so let's take a look, we have first of all 3 coordinate axis, the x axis, the y axis and the z axis. And these axes are actually themselves create planes, there's the xy plane which I've drawn in red here, let me label that xy plane now every point on the xy plane has a z coordinate of 0 so it's also called z=0 that's an identifying characteristic for the plane so that's an equation for the plane. And there's also the in purple here I've got the yz plane, yz plane is also called x=0 because every point on it has x coordinate 0.
And finally there's the xz plane which is in blue here so I'll write that down here xz plane and of course the equation for that is y=0 and you probably notice the way to remember what the equation of the plane is, is what letter is missing? xz so the missing letter is y, so y=0 is the equation. Now what's really important to know about the coordinate plane is that these are called is that they divide 3 dimensional space into 8 portions which are called octants right just like the xy plane is divided into quadrants by the x and y axis these coordinate planes divide space into 8 regions and they're called octants. And we often talk about the first octant but the other ones aren't really numbered the first octant is the one where all the coordinates xy and z are positive.
Now I want to talk about planes in a little more depth and I can't quite get more planes drawn onto this picture so let's take a look at a graphing program that I have on my computer. Okay here we are in grapher which is a computer that I have on my Mac and there's a coordinate system drawn 3 dimensional coordinate system I have my cursor on the x axis, this is the y axis and this is the z axis. And I can show you really quickly that's z equals 0 right that's the xy plane and y equals 0 that's the xz plane ad x=0 that's the yz plane and if you want to see them all at once now you can see that the octant that's facing us, this is the first octant that I was speaking of and this octant all the coordinates x, y and z are positive.
Now let's take a look at some other planes, z=2, z=2 you can tell its identifying characteristic is that the z values are all 2 and you'll also notice that it's parallel to z=0, it's parallel to the xy plane. In fact z equals any constant, it's going parallel to the xy plane, here's y equals negative 3, y equals negative of course it's going to be parallel to y=0 the xz plane. And here's y=x in 2 dimensions in the xy plane this is just a line but in 3 dimensions this is a plane. So the defining characteristic y=x tells you that every point on this plane has the same x any coordinate. So there are lots and lots of other kinds of planes that we'll be talking about in the near future but it's important to know simple equations like z=2 or like y equals negative 3 describe planes.