Parametric forms for lines and vectors
In many situations, it is useful to have an alternative way of describing a curve besides having an equation for it in the plane. A parametric form for a line occurs when we consider a particle moving along it in a way that depends on a parameter , which might be thought of as time. Thus both and become functions of . The simplest parameterisation are linear ones.
In this step we
look at parameterisations of the simplest curves: lines
show how to go from parametric equations to Cartesian equations
Parameterisations of a line
A simple example of a parameterisation is a linear parameterisation:
As varies, the point moves along a line. When , , when , , and when , . Clearly also fractional values of are allowed.
How do we go from a parametric form for a line to one of the more usual Cartesian forms involving and ? We eliminate the parameter . For example for the parametric equation we have
so that or . Please check that this is the same line – with a different equation.
Q1 (E): Find a Cartesian equation for the parametric line .
How to go from a Cartesian equation to a parametric form?
On the other hand, how can we go from a Cartesian equation of a line, say to a parametric form? It is important to realise that although the Cartesian form is more or less unique, this is not at all the case for a parametric form.
One way is to introduce a parameter in a simple way, say by setting . Then the original equation gives . Putting these together gives the parameterisation .
Q2 (E): Find another parameterisation of by setting .
Using vectors to parameterise lines
The rest of this step is about vectors, for those who are familiar with them. A vector is a directed line segment, or geometrically a difference between two points in the plane. We will denote vectors by round brackets, such as . This represents a relative displacement of to the right in the -direction and up in the -direction. This plays a similar role to the notion of slope, which is the ratio of the relative displacement of the and -directions or “rise over run”. Using vectors allows us to be more precise about the notion of direction of a line.
We agree that vectors are not fixed in place, as they are only determined in a relative sense.
Suppose that we consider the line through the point which goes in the direction . This line can be expressed as
which is now in parametric form with parameter .
What is the Cartesian equation of this line? We may eliminate from the equations , to get so the equation is .
Now imagine that we would like to traverse the same line, only faster. To move through the same line at double the speed, we replace with :
Notice that the direction vector has doubled, but the Cartesian equation of this line remains unchanged. We can also move backwards along the line by relpacing with :
Now the direction vector has been negated, but again the Cartesian equation remains unchanged.
So we see that the parameterisation of a curve describes more than the Cartesian equation, a parameterisation describes how we traverse the curve: fast or slow, forwards or backwards.
A1. This is also .
A2. Another parameterisation is .
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