Vector Spaces in Physics 8/25/2015 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 Figure 1-1. Where is the 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: displacement from r12 point 1 to point 2 (1-1) distance, angle measured counterclockwise from due East (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. 1 - 1 Vector Spaces in Physics 8/25/2015 distance point 2 r angle east point 1 Figure 1-2. A vector, specified by giving a distance and an angle. 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: rP displacement from point O to point P (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 r12 r2 r1 (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 and be two displacement vectors, each defined by giving its length and angle: A (A, ), A (1-4) B (B, B ). 1 - 2 Vector Spaces in Physics 8/25/2015 Here we follow the convention of using the quantity A (without an arrow over it) to represent the magnitude of ; and, as stated above, angles are measured counterclockwise from the easterly direction. Now imagine points 1, 2, and 3 such that represents the displacement from 1 to 2, and represents the displacement from 2 to 3. This is illustrated in figure 1-3. 3 B 2 B A A 1 Figure 1-3. Successive displacements and . Definition: The sum of two given displacements and is the third displacement C which has the same effect as making displacements and in succession. It is clear that the sum exists, and we know how to find it. An example is shown in figure 1-4 with two given vectors and and their sum . It is fairly clear that the length and angle of 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 and B be the two vectors shown in figure 1-4: =(10 m, 48), B =(14 m, 20). Determine the magnitude and angle from due east of their A B sum C , where C A B . 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 1 - 3 Vector Spaces in Physics 8/25/2015 C A B 48-20 3 B = 28 3 14 m 3 48 2 20 2 20 A 2 C 10 m 48 180-28 1 C = 152 1 1 Figure 1-4. Example of vector addition. Each vector's direction is measured counterclockwise from due East. Vector is a displacement of 10 m at an angle of 48and vector is a displacement of 14 m at an angle of 20. cosines: 2 2 2 C = A + B -2AB cos 2 giving 2 2 1/2 C = [(10 m) + (14 m) - 2(10m)(14m)cos 152] = 23.3072 m . The angle 1 can be calculated from the law of sines: sin 1 / B = sin 2 / C giving -1 1 = sin .28200 = 16.380 . The angle C is then equal to 48 - 1 = 31.620 . The result is thus C 23.3072 m, 31.620 . One conclusionA toB 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: A B AB cos(B -A ) (1-5) 1 - 4 Vector Spaces in Physics 8/25/2015 This simple definition is illustrated in figure 1-5. One special property of the dot product is its relation to the length A of a vector : B 2 A A A (1-6) B - A This in fact can be taken as the definition of the length, or magnitude, A. A An interesting and useful type of vector is a unit vector, defined as a vector of length 1. Figure 1-5. Vectors and . Their dot We usually write the symbol for a unit vector product is equal to A B cos( B - A). with a hat over it instead of a vector symbol: Their cross product A B has magnitude uˆ . Unit vectors always have the property A B sin(B - A), directed out of the paper. uˆ uˆ 1 (1-7) Another use of the dot product is to define orthogonality of two vectors. If the angle between the two vectors is 90, they are usually said to be perpendicular, or orthogonal. Since the cosine of 90 is equal to zero, we can equally well define orthogonality of two vectors using the dot product: ABAB 0 (1-8) Example. One use of the scalar product in physics is in calculating the work done by a force F acting on a body while it is displaced by the vector displacement d . The work done depends on the distance moved, but in a special way which projects out the distance moved in the direction of the force: Work Fd (1-9) Similarly, the power produced by the motion of an applied force whose point of application is moving with velocity v is given by Power Fv (1-10) In both of these cases, two vectors are combined to produce a scalar. Example.A B To find the component of a vector A in a direction given by the unit vector nˆ , take the dot product of the two vectors. Component of A along n A nˆ (1-11) The Vector Product. The vector product, or cross product, is considerably more complicated than the scalar product. It involves the concept of left and right, which has an interesting history in physics. Suppose you are trying to explain to someone on a distant planet which side of the road we drive on in the USA, so that they could build a car, bring it to San Francisco and drive around. Until the 1930's, it was thought that there was no way to do this without referring to specific objects which we arbitrarily designate as left-handed or right-handed.