As you know, two pieces of information determine the equation of a line -- for example, two points on the line, or a point on the line and the slope. Here we introduce a new way of describing lines using vectors and parameterizations. This description, unlike our previous methods, can be generalized to lines in space.

Consider the line through the two points (1, 2) and (3, 3). To describe this line using vectors, we first compute the displacement vector *v* = <2, 1>, which has its tail at (1, 2) and its head at (3, 3). This vector describes the direction of the line. The picture below (left) shows the line we seek together with this displacement vector.

Next notice that we can now produce any point on the line by adding multiples of the direction vector <2, 1> to the vector <1, 2>. In the right figure below, we illustrate how it works by adding two copies of <2, 1> to <1, 2>. In this way, we see that the point (5, 4) is on the line, because <5, 4> = <1, 2> + 2<2, 1>.

**Figure 3.1.** (Left) Adding the displacement vector <2, 1> to <1, 2> yields the vector <3, 3>.

(Right) Starting at (1, 2) and adding two displacements of <2, 1> will take us twice as far along the line.

More generally, note that any point on the line can be described by <1,2> + *t* <2, 1>, for some appropriate value of *t*. Negative values of *t* will correspond to points on the line heading in the other direction. Expanding the expression results in a parametric description of the line:

<1, 2> + *t* <2, 1> = <1, 2> + <2*t*, *t*> = <1 + 2*t*, 2 + *t*>.

That is, the equations *x*(*t*) = 1 + 2*t* and *y*(*t*) = 2 + *t* parameterize the line. Notice that setting

*t* = 0 produces the starting point (*x*(0), *y*(0)) = (1, 2), and if *t* = 1 we get (*x*(1), *y*(1)) = (3, 3). Hence we let *t* vary from *t* = 0 to *t* =1 to trace out the line segment from (1, 2) to (3, 3).

Parametric Plotter

**Exercise 3.1**
Use what you have just learned to find the parameterization of the directed line segment from (0,4) to (3,0). Since (0, 4) is your starting point, your parameterization (*x*(*t*), *y*(*t*)) should satisfy (*x*(0), *y*(0)) = (0, 4).

**Exercise 3.2**

Produce a plot of a right triangle having sides of length 3, 4, and 5.