ectors The word ector comes from the Latin word ectus which means carried. It is best to think of a ector as the displacement from an initial point P to a terminal point Q. Such a ector is expressed as PQ. The figure below shows a typical ector. Notice that the ector appears as a line with an arrow at one end. This is referred to as a directed line segment. The end with the arrow is the head of the ector, or the terminal point, while the other end is the starting, or initial point. Q Vectors hae two important aspects: direction and magnitude. The direction tells you where to point the ector and the magnitude tells you how far to go in that direction. The magnitude of the ector PQ is written as PQ. Note that while this looks just like we take the absolute alue of the ector PQ, it means the distance between points P and Q. And so, to compute the magnitude of a ector, we use the distance formula. We use two P different formulas, depending on whether or not the ector is in two dimensions or three dimensions. If our points are in two dimensions, then P can be expressed as (p 1, p 2 ) and Q can be expressed as (q 1, q 2 ). We hae that PQ can be written as q1 p1, q2 p2 and PQ ( q p ) ( q p ). 1 1 It helps to think of the magnitude of a ector as the hypotenuse of a triangle. Let us consider the aboe ector placed on the coordinate axes. q 2 p 2 P p 1 q 1 Figure 1: A Vector Q Notice that the base of the triangle (which also happens to be parallel to the x-axis) is q1 p1 while the height of the triangle (parallel to the y-axis) is q 2 p 2. To find the length of the hypotenuse, we would square the two sides, add them together, and take the square root, just as we did aboe. This leads to another point. Aboe we wrote the ector PQ as q1 p1, q2 p2. When expressing a ector, we break it down into components along the coordinate axes, in this case the x and y axes. 1
In passing, we note that two ectors are equal if each component is the same. That is, a, b c, d if and only if a c and b d. Example 1: Suppose P has coordinates (1, 1) and Q has coordinates (2, 3). Find the ector and calculate its magnitude. PQ PQ 2 1,3 1 1,2. The magnitude of PQ is PQ 1 2 5. If our points are in three dimensions, then P can be expressed as (p 1, p 2, p 3 ) and Q can be expressed as (q 1, q q 3 ). Using similar reasoning as in the two-dimensional case, we 2, hae that the ector PQ can be written as q1 p1, q2 p2, q3 p3 and its magnitude is 2 gien by PQ ( q p ) ( q p ) ( q p ). 1 1 3 3 Example 2: Suppose P has coordinates (0, 1, 1) and Q has coordinates (1, 2, -1). Find the ector PQ and calculate its magnitude. PQ 1 0, 2 1, 1 1 1,1, 2 2 PQ 1 1 ( 2) 6.. The magnitude of PQ is gien by For most of our purposes, we shall consider ectors in three dimensions. Many applications in physics, howeer, use only two dimensions. As we can see with the formulas aboe, going from three dimensions down to two dimensions is rather simple: just pretend that the last component is not there and use the same formula. Often times when dealing with ectors, it is necessary to add them. When adding two ectors, we add the components. And when we subtract two ectors, we subtract the components. That is, 2
Vector Addition and Subtraction If 1, 2, 3 and w w1, w2, w3, then w w, w, w 1 1 3 3 w w, w, w 1 1 3 3 If and w only hae two components, then you only add the two components, ignoring the third one in the formula aboe. Geometrically, one can draw a parallelogram to see how to add ectors. Basically, you just stick the tail of one ector at the head of the other ector. Then you draw a ector from the tail of the first ector to the head of the second ector. w +w w Figure 2: Vector Addition Example 3: Add the ectors 0,1, 1 and w 3, 4, 2. We hae that w 0 3,1 4, 1 2 3,5,1. There is no such thing as the multiplication of two ectors. Later on, we will talk about topics that are similar to multiplication, but that is another matter entirely. We can, howeer, scale each of the components of a ector u by a multiple k. Essentially, we are adding k copies of the ector u. This resulting ector, ku is called the scalar product. If k is positie, the ector ku points in the same direction as u and if k is negatie, then the ector ku points in the opposite direction. 3
Scalar Multiplication If 1, 2, 3 and k is a real number, then the scalar product k is gien by k k1, k2, k3 If only has two components, the same formula holds, but we just ignore the third component. Example 4: Suppose u 1, 3, 0 and 2,1, 4. Compute the following quantities: (i) 2u (ii) u + 3 (iii) 3u 2 (i) 2u 2 1,2 ( 3),2 0 2, 6,0 (ii) u 3 1, 3,0 32,31,34 1, 3,0 6,3,12 7,0,12 (iii) 3u 2 3 1,3 ( 3),3 0 2 2,2 1,2 4 3, 9,0 4,2,8 1, 11, 8. There are some common ectors in three-dimensions that come up so often, they hae been gien a special notation. They are i 1, 0, 0, j 0,1, 0, and k 0,0,1. Notice that each one corresponds to a one unit displacement in each direction, x, y, and z, respectiely. We can use these ectors to decompose a ector into its x, y, and z components. We called this form the component form of the ector. Example 5: Write the ector 3, 2, 6 in component form. 3, 2, 6 3,0,0 0, 2,0 0,0,6 3 1,0,0 2 0,1,0 6 0,0,1 3i 2j 6k 4
In general, the component form is gien just by taking the entries of the ector and attaching i, j, and k respectiely to the alues. We shall use this component form from this point on. It is possible for two ectors to point in the same direction but hae different lengths. Indeed, one is just a scalar multiple of the other. It would be nice if we could specify the ector which points in a certain direction. We can do that, though, if we understand that the ector we are referring to has a magnitude of 1. Such ectors are called unit ector. Unit Vector A unit ector u in the same direction as the ector is gien by u, proided 0. The ector u has the property that u 1. Example 6: Express the ector 3i 2j 6k as a unit ector. To express as a unit ector, we need to first determine its length. 2 ( 3) 2 ( 6) 9 4 36 49 7. 1 3 2 6 j k i j k. 7 7 7 7 Thus, we hae that u 3i 2 6 5