Vector Spaces in Physics 8/22/2014 1 - 1 Chapter 1. Vectors We are all familiar with the distinction between things which have a direction and those which don't. The velocity of the wind (see figure 1.1) is a classical example of a vector quantity. There are many more of interest in physics, and in this and subsequent chapters we will try to exhibit the fundamental properties of vectors. Vectors are intimately related to the very nature of space. Euclidian geometry (plane and spherical geometry) was an early way of describing space. All the basic concepts of Euclidian geometry can 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. The modern use of Cartesian vectors provides the mathematical basis for much of physics. A. The Displacement Vector The preceding discussion did not lead to a definition of a vector. But you can 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 point in space to another. Assuming we know how to measure distances and angles, we can define a displacement vector (in two dimensions) in terms of a distance (its magnitude), and an angle:  12displacement frompoint 1 to point 2distance, angle measured counterclockwise from due Eastr(1-1) (See figure 1.2.) 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 vectors. Figure 1-1. Where is the vector?Vector Spaces in Physics 8/22/2014 1 - 2 B. Vector Addition 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 point 1 to point 2 should be expressed in terms of the position vectors for 1 and 2. We are be tempted to write 1212rrr (1-3) A "difference law" like this is certainly valid for temperatures, or even for distances along a road, if 1 and 2 are 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. Definition of 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. 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: ).,(),,(BABBAA (1-4) point 1 point 2 r east angle distance Figure 1-2. A vector, specified by giving a distance and an angle.Vector Spaces in Physics 8/22/2014 1 - 3 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 1-4 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. The example below illustrates this calculation. Example: Let A and B be the two vectors shown in figure 1-4: A=(10 m, 48), B=(14 m, 20). Determine the magnitude and angle from due east of their sum C, where BAC. The angle opposite side C can be calculated as shown in figure 1-4; the result is that 2 = 152. Then the length of side C can be calculated from the law of AB123ABFigure 1-3. Successive displacements A and B.Vector Spaces in Physics 8/22/2014 1 - 4 cosines: C2 = A2 + B2 -2AB cos 2 giving C = [(10 m)2 + (14 m)2 - 2(10m)(14m)cos 152]1/2 = 23.3072 m . The angle 1 can be calculated from the law of sines: sin 1 / B = sin 2 / C giving 1 = sin-1 .28200 = 16.380 . The angle C is then equal to 48  - 1 = 31.620 . The result is thus  23.3072 m, 31.620C . One conclusion to be drawn from the previous example is that calculations using the geometrical representation of vectors can be complicated and tedious. We will soon see that the component representation for vectors simplifies things a great deal. C. Product of Two Vectors Multiplying two scalars together is a familiar and useful operation. Can we do the same thing with vectors? Vectors are more complicated than scalars, but there are two useful ways of defining a vector product. The Scalar Product. The scalar product, or dot product, combines two vectors to give a scalar: )-(cABosBABA -