 Hello, and welcome to this screencast on section 9.5, Lines and Planes in Space. This screencast is going to cover lines in space. You're likely familiar with the slope intercept definition of a line. This definition works for two dimensions. We'd like to extend the definition of a line to three or higher dimensions, however, the notion of slope, often referred to as rise over run, isn't well defined for three or higher dimensions. Since the slope tells us the direction of a line, an alternate way to think about slope is using a vector, since vectors tell us a direction. Points on the line can then be described by the sum of two vectors. The first vector traveling from the origin to some point P, and the second vector traveling along some scalar multiple of the vector V, which tells us the direction of the line. This idea leads to a new definition of a line. A line in space is the set of terminal points of vectors emanating from some given point P that are parallel to a fixed vector V. The vector form of a line through a point P in the direction of the vector V is given by the following expression, which is a sum of two vectors, as we saw in the last slide. The first vector is a vector from the origin to some point P on the line. The second is a scalar multiple of the vector V. The vector V is often referred to as the direction vector of a line. And I want to note here that this definition works for any point P on the line, not just the y-intercept as the previous slide suggested. Shown here is an example of a line, the vector definition of a line, that we can reach any point on this line by first traveling from the origin to the point P, then traveling along a scalar multiple of the vector V. We can reach points anywhere on this line by choosing the value of the scalar T appropriately. These pictures here show how three different values of the scalar T puts us at different points along the line. Thus far, this has all been pictured in two dimensions, but all of these ideas extend to three dimensions or higher. We now wish to look at an alternate way to define a line in space. Consider the vector form of a line R of T in three space through the point P with coordinates x-naught, y-naught, and z-naught, with direction vector V that has components a, b, and c. Using vector addition, we can use the coordinates from the point P and the components of the vector V to describe the line in terms of a single vector. And from this last vector that we have here, we get three equations that describe the x, y, and z-coordinates of points on the line in terms of the variable T. These equations are called the parametric equations of a line. The variable T represents an arbitrary scalar that is called the parameter. Note that there are many different parametric equations for the same line. For example, we could choose another point P on the line, or we could choose a different direction vector, and these would produce another set of parametric equations that describe the same line. It will sometimes be useful to think of T as a time parameter in the parametric equations as telling us where we are on the line at each time. In this way, the parametric equations describe a particular walk along the line of which there are many different ways that we could walk a line.