Vector Spaces in Physics 8/11/2010 1 - 1 Chapter 1. Vectors Everyone is familiar with the distinction between things which have a direction and those which don't. When you are asked, "Which way is the wind blowing?" you have to point; when you are asked, "What is the temperature," you don't – and in fact, you can't. The direction of the temperature just doesn't have any meaning. Consider the following list of important and interesting quantities: 1. The price of a ticket to a baseball game. 2. The direction to San Jose. 3. The processor speed of a Mac G4. 4. The depth of the ocean under the Golden Gate Bridge. 5. The love of a child for her dog, 6. How windy it is at Pac Bell park. 7. The location of the student union. Are any of these things vectors? Which ones? To refine the concept of vector, we have to separate physical measurement from sociological context. Here are the physical measurements corresponding to the quantities just listed: 1. Number of dollars paid for a ticket to a baseball game. 2. Compass bearing to follow to go to San Jose. 3. Cycles of Mac G4 clock signal per second. 4. Depth of water under center of Bridge span. 5. No scientific definition (does this mean it's not important?) 6. Speed of wind, compass direction it comes from. 7. Distance and direction to the student union. All but one of these things can be measured with instruments. Some involve direction, some don't. (What about the water depth?) Are any of these quantities vectors? Vectors have something to do with the nature of space. Euclidian geometry (plane and spherical geometry) was a very early way of describing space precisely. All the basic concepts of Euclidian geometry can be expressed in terms of angles and distances. A number of the quantities above could also be expressed in terms of angles and distances. A more recent development in describing space was the introduction by Descartes of coordinates along three orthogonal axes. It is often said that "A vector is a quantity with both magnitude and direction." Is this a complete definition? A. The Displacement Vector The preceding discussion did not lead to a definition of a vector. But you may be able to convince yourself that all of the things we think of as vectors can be related to a single fundamental quantity, the vector r∆ representing the displacement from one given point to another. Quantities with an arrow over them will represent vectors for us. Assuming Figure 1-1. Where is the vector?Vector Spaces in Physics 8/11/2010 1 - 2 we know how to measure distances and angles, we can define a displacement vector (in two dimensions) in terms of a distance and an angle: ( )12displacement frompoint 1 to point 2distance, angle measured counterclockwise from due Eastr ∆ =  ≡(1-1) Note that to a given pair of points corresponds a unique displacement, but a given displacement can link many different pairs of points. Thus the fundamental definition of a displacement gives just its magnitude and angle. We will use the definition above to discuss certain properties of vectors from a strictly geometrical point of view. Later we will adopt the coordinate representation of vectors for a more general and somewhat more abstract discussion of vector spaces. B. Vector Addition and Other Operations A quantity related to the displacement vector is the position vector for a point. Positions are not absolute – they must be measured relative to a reference point. If we call this point O (the "origin"), then the position vector for point P can be defined as follows: {}displacement from point O to point PPr ≡ (1-2) It seems reasonable that the displacement from 1 to 2 can be expressed in terms of the position vectors for 1 and 2. We would be tempted to write 1212rrr−=∆ (1-3) This "difference law" would certainly be true for temperatures, or even distances along a road, if 1 and 2 were two points on the road. But what does subtraction mean for vectors? Do you subtract the lengths and angles, or what? When are two vectors equal? In order to answer these questions we need to systematically develop the algebraic properties of vectors. We will let A, B, C, etc. represent vectors; for the moment, the only vector quantities we have defined are displacements in space. Other vector quantities which we will define later will obey the same rules. 1. Equality. Two vectors are said to be equal if they have the same magnitude and angle. point 1 point 2 r∆ east angle distance Figure 1-2. A vector in terms of distance and angle.Vector Spaces in Physics 8/11/2010 1 - 3 2. Vector Addition. The sum of two vector displacements can be defined so as to agree with our intuitive notions of displacements in space. We will define the sum of two displacements as the single displacement which has the same effect as carrying out the two individual displacements, one after the other. (See Figure (1-3).) To use this definition, we need to be able to calculate the magnitude and angle of the sum vector. This is straightforward using the laws of plane geometry. (The laws of geometry become more complicated in three dimensions, where the coordinate representation is more convenient.) Let A and B be two displacement vectors, each defined by giving its length and angle: ).,(),,(BABBAAθθ== (1-4) Here we follow the convention of using the quantity A (without an arrow over it) to represent the magnitude of A; and, as stated above, angles are measured counterclockwise from the easterly direction. Now imagine points 1, 2, and 3 such that A represents the displacement from 1 to 2, and B represents the displacement from 2 to 3. This is illustrated in figure 1-3. Definition: The sum of two given displacements A and B is the third displacement C which has the same effect as making displacements A and B in succession. It is clear that the sum C exists, and we know how to find it. An example is shown in figure 2 with two given vectors A and B and their sum C. It is fairly clear that the length and angle of C can be determined (using trigonometry), since for the triangle 1-2-3, two sides and the included angle are known. θAθB123ABFigure 1-3. Successive displacements A and B. 48° 20° 1 2 3 ABθC 10 m 14 m CBAC+=θ1