In this section we need to take a look at the equation of a line in R 3 R 3 . As we saw in the previous section the equation y = m x + b y = m x + b does not describe a line in R 3 R 3 , instead it describes a plane. This doesn’t mean however that we can’t write down an equation for a line in 3-D space. We’re just going to need a new way of writing down the equation of a curve. So, before we get into the equations of lines we first need to briefly look at vector functions. We’re going to take a more in depth look at vector functions later. At this point all that we need to worry about is notational issues and how they can be used to give the equation of a curve. The best way to get an idea of what a vector function is and what its graph looks like is to look at an example. So, consider the following vector function. → r ( t ) = ⟨ t , 1 ⟩ r → ( t ) = ⟨ t , 1 ⟩ A vector function is a function that takes one or more variables, one in this case, and returns a...
We’ll start the chapter off with a fairly short discussion introducing the 3-D coordinate system and the conventions that we’ll be using. We will also take a brief look at how the different coordinate systems can change the graph of an equation. Let’s first get some basic notation out of the way. The 3-D coordinate system is often denoted by R 3 R 3 . Likewise, the 2-D coordinate system is often denoted by R 2 R 2 and the 1-D coordinate system is denoted by R R . Also, as you might have guessed then a general n n dimensional coordinate system is often denoted by R n R n . Next, let’s take a quick look at the basic coordinate system. This is the standard placement of the axes in this class. It is assumed that only the positive directions are shown by the axes. If we need the negative axes for any reason we will put them in as needed. Also note the various points on this sketch. The point P P is the general point sitting...