1.1.4 Directed Lines and Line Segments
In Linear Algebra I taught you that a line through the origin is a one dimensional subspace. One dimensional, meaning it could be represented as all linear combinations of a single (non-zero) vector. Let's think about that in terms of parameterized curves:
Suppose is a non-zero vector in . Then the line through the origin parallel to is the image of the vector function:
We can represent this parametrically as:
In Multi we will be concerned with lines that may not pass through the origin. We can accomplish this by simply shifting a line through the origin by adding a fixed vector.
Hence, the general form for a line in is:
where the vector is a fixed vector in the plane.
Experiment with the applet below. You can change the slope vector or the fixed point. Animate will show you how the line is traced out as the parameter increases from 0 to 1.
It's important to remember that when we analyze the curve that results from some path, we are analyzing the image of that path. While the parameterization I gave you above is one way to parameterize a directed line, there are many others. It is not a requirement that the component functions be linear! For example, show that the image curve of the following path is a segment of a directed line: What is the slope of the line? If is a function whose graph is the image curve of , what is the domain of ?
The procedure I modeled in the last question to describe the image curve as is called eliminating the parameter and the resulting equation directly relating the variables and is called a rectangular equation (rectangular because the grid created by the and axes divides the plane into lots of little rectangles).