 Hello, and welcome to this screencast on section 9.2, Vectors. This screencast is going to cover representation of vectors. A vector is a quantity that possesses the attributes of magnitude, or length, and direction. We can represent a vector geometrically as a directed line segment, with the magnitude as the length of the segment, and an arrowhead indicating the direction. According to the definition, a vector possesses the attributes of magnitude and direction. The vector's position, however, is not mentioned. Consequently, we regard any two vectors having the same magnitude and direction as equal. For example, these vectors all have the same magnitude and direction. Therefore, the directed line segments all represent the same vector. This means the same vector can be drawn in the plane in many different ways. Let's look at some ways to represent vectors. On the left graph, we have the vector 3,2, which represents a horizontal change of 3 units and a vertical change of 2 units. We say that the x component of this vector is 3, and the y component is 2. This vector is drawn in standard position, which means the tail of the vector is placed at the origin. Here, we've named this vector v, and you'll notice we use boldface letters to represent vectors. We also use pointed brackets for vectors to distinguish them from points. On the right side, we have another way to represent vectors. Given two points p and q, we can obtain the vector from p to q by placing the tail end of the vector at p and the tip of the vector at q. When we use points to define vectors, we often refer to that vector using the names of the points, p and q, with an arrow over top. Note that this vector from p to q has x component 3 and y component 2, which is the same as the vector drawn in standard position on the left graph. So, the two vectors represented on this page are equal. Note that another way to get the components of a vector obtained by points is to take the difference of the coordinates of the points. So here, to get the x component of the vector from p to q, we would take the x coordinate of q minus the x coordinate of p. Similarly, to get the y component, we take the y coordinate of q minus the y coordinate of p. Be careful, though, as the order that you take the difference determines the direction of your vector. One other way to define a vector is using a single point. Given some point p, we will frequently consider the vector from the origin to p. We use the same notation as before when we define vectors between two points using the letter capital O for the origin. Vectors represented this way are called the position vector of the point p. We are going to take an opportunity here to quickly introduce some notation. The magnitude or length of a vector v is denoted using vertical bars that look like the familiar absolute value notation. We can calculate the magnitude of a vector using the distance formula. For example, the magnitude of the vector from the origin to p is the square root of 13. That is, the vector shown here has length equal to the square root of 13. You will work more with the magnitude of vectors in the activities in section 9.2. While we've been working with vectors in two dimensions, note that all the work we've done so far also holds for vectors in three dimensions. In addition to considering the x component and y component in three dimensions, we also need to consider the z component of vectors. Pictured here is the vector with x component 2, y component 4, and z component equal to 3. One other way that we can represent vectors is using the standard unit vectors, which are important vectors in the physical sciences. Each of these vectors represents a one unit change in a specific direction. The vector i represents a one unit change in the x direction, the vector j in the y direction, and the vector k in the z direction. Given a vector with components a, b, and c, we can represent this as a sum of the standard unit vectors. Note that this representation relies on vector addition, which you will learn more about in section 9.2. As a quick example, we can represent the vector 2, 4, 3 as 2i plus 4j plus 3k.