At this point you should be very familiar with graphing linear equations of the form , where is the slope of the line and is the -intercept. Unfortunately, there is no equivalent way of representing lines in , or better yet, . In this section we will develop an alternative way of representing lines.
The point , corresponding to each value of is the location of the ladybug in the coordinate plane at time .
It appears that the points corresponding to lie on a line with slope . In the next problem we will find the equation of the line that contains the path of the ladybug.
One approach to finding the equation is to solve one of the given equations for , then substitute into the other equation. Solving for gives us Substituting this expression into we get Plotting the line together with the path we plotted earlier, we see that our original path lies on the line.
If we do not restrict to values between 0 and 3, we can get every point on the line .
Equations such as
are called parametric equations, and is called a parameter.When given an equation of the form , we recognize it as an equation whose graph is a line and we don’t need to make a table of values to sketch the graph of the equation. We should be able to do the same for parametric equations of lines. In the next Exploration we will examine our equations carefully to see if we can discern any patterns that would help us plot the line without making a table of values.
Consider the table of values we constructed in Exploration init:paramline2d.
Observe that every time increases by , increases by and decreases by . An increase by in the -coordinate combined with a decrease of in the -coordinate corresponds to the coefficients and of in the parametric equations. Our findings are in agreement with the fact that the slope of the line is .
We will capture the “rise” and “run” aspect of the line by using vector . Sketching together with the line, we find that the vector is parallel to the line.
Vector is called a direction vector. Observe that the components of are the coefficients of in the parametric equations.
Next, let’s turn our attention to constants and . They are the values of and when , and correspond to the point .
Recall that in Exploration init:paramline2d we said that the given parametric equations describe the position of a ladybug crawling in the coordinate plane. If we are concerned with the position of the ladybug at time , the point is very important - this is where the bug is located when . But, if we are using parametric equations simply to describe the line , without regard for when the bug is located at each point, then the point is not any more special than any other point on the line. In fact, we can use any other point on this line to find another set of parametric equations that describe the same line. In Practice Problem prob:paramnotunique you will be asked to show that equations
describe the same line. Thus, parametric representations are not unique.We can generalize our observations in Exploration init:paramline2dpart3 as follows.
Observe that every time increases by , increases by , increases by , and increases by . The pattern “out , over , up ” is illustrated in the diagram.
We will use a direction vector to represent the direction of the line. Note that the components of are the same as the coefficients of in the parametric equations.
Also, observe that the constants correspond to the coordinates of the point on the line for which .
As before, the point may be replaced with any other point on the line to produce a different set of parametric equations describing the same line.We can generalize our observations in Exploration init:paramline3d as follows.
After working with parametric equations of lines in and , it should be easy to surmise what parametric equations of lines look like for lines in .
Consider again the parametric equations from Exploration init:paramline2d.
These equations describe the line by providing coordinates of individual points on the line, as shown in the left figure below.
If we consider the points on the line to be heads of vectors, then we can think of the line as being “traced” by a vector as changes (See above, right). We can capture this algebraically by using a vector equation for the line: Note that the direction vector and the point , now vector , are easy to identify in the equation.
We can use vector equations to describe lines in using the following formula.