Equations of Lines
In this section we need to take a look at the equation of a line in
. As we saw
in the previous section the equation
does not describe
a line in
, instead it describes a plane. This doesn’t mean however that we
can’t write down an equation for a line in 3D 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.
A vector function is a function that takes one or more variables, one in this case, and
returns a vector. Note as well that a vector function can be a function of two or more
variables. However, in those cases the graph may no longer be a curve in space.
The vector that the function gives can be a vector in whatever dimension we need it to
be. In the example above it returns a vector in
. When we get to the real
subject of this section, equations of lines, we’ll be using a vector function that returns
a vector in
Now, we want to determine the graph of the vector function above. In order to find
the graph of our function we’ll think of the vector that the vector function returns as a
position vector for points on the graph. Recall that a position vector, say
, is a vector that starts at the origin and ends at the point
.
So, to get the graph of a vector function all we need to do is plug in some values of
the variable and then plot the point that corresponds to each position vector we get out
of the function and play connect the dots. Here are some evaluations for our example.
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View Full DocumentSo, each of these are position vectors representing points on the graph of our vector
function. The points,
are all points that lie on the graph of our vector function.
If we do some more evaluations and plot all the points we get the following sketch.
In this sketch we’ve included the position vector (in gray and dashed) for several
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 Fall '08
 prellis
 Equations, Parametric equation, Vectorvalued function

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