We live in a world of three spatial dimensions and one time dimension. In our ordinary experience we find that these four dimensions are all independent of each other. We can imagine a reference frame as a set of perpendicular coordinates and a clock. We can measure the velocity of something moving with respect to this first reference frame. We can also imagine a second reference frame moving at a constant velocity with respect to the first. The velocity of that same object as measured in the second reference frame turns out to be given by the vector addition of the velocity of the object as measured in the first reference frame and the velocity of the first reference frame with respect to the second reference frame. This is simply a fancy way of saying that if you walk forward on a moving bus, your velocity with respect to the ground is your velocity as measured on the bus plus the velocity of the bus with respect to the ground. This behavior is referred to as Galilean relativity. Space and time “behave” like “they are supposed to behave.” Time is totally independent of the spatial dimensions and each spatial dimension is independent of the other spatial dimensions. (Recall that we used this notion of Galilean space when we asserted that there were three independent ways atoms could move—and have energy—when we predicted heat capacities using the Particle Model of Thermal Energy.)
Most phenomena we encounter in our everyday experience is consistent with the Galilean model of space-time described in the previous paragraph. However, the motion of the electrons moving from the back of your color television picture tube and striking the phosphorescent material on the inside front surface producing the nice color images we like to watch, can not be described by the common-sense notions of the Galilean space-time model! The motion must be described using special relativity. In the special-relativity model of space-time the three spatial dimensions are not independent of time. Time and space get mixed together. The electron—if it had a clock with it—would think it got from the back of the picture tube to the front in a much shorter time than you would measure. And you and the electron would both be correct! The clocks are not messed up. They are giving the correct times. It is just that the times really are different! These kinds of seemingly weird behaviors are really the way space-time “works” when relative velocities get large, meaning appreciable compared to the speed of light, which is \(3 \times 10^8 ~m/s\). What we think of as normal space-time behavior—Galilean space-time—works only for speeds slow compared to the speed of light.
Galilean space-time also breaks down when we get near really large masses, like black holes.
Except for a little digression in Part 3 when we try to make sense of the origin of magnetism, we will assume we can use the Galilean space-time model. (Magnetism is a direct manifestation of the effects of special relativity, even when there are no material objects moving at fast speeds. If the speed of light—and other electromagnetic radiation—were infinite, there would be no magnetism.) So, even though we won’t pursue these “weird” effects further, it is useful to keep in mind the limitations to our understanding of space-time as we normally experience it and described in the Galilean Space-Time Model.
Describing Motion, Force, and Other Interesting Things in Galilean Space- Time
What we are really interested in now is how we describe how objects move in Galilean space-time. (We will omit the phrase Galilean space-time from now on, simply assuming that we are always using this space-time model.) How do we describe and make sense of “things” like velocity, force, momentum and torque that have directional properties as well as a magnitude?
Basic Vector Properties
A very useful idea (we frequently use the word “construct” for an idea like this) is the concept of a vector. We can give a definition of a vector as something that exhibits both directional properties as well as a magnitude. But beware! The quote below points out a trap that is easy to fall into:
"It often does more harm than good to force definitions on things we don't understand. Besides, only in logic and mathematics do definitions ever capture concepts perfectly. The things we deal with in practical life are usually too complicated to be represented by neat, compact expressions. … In any case, one must not mistake defining things for knowing what they are."
Marvin Minsky, from The Society Of Mind, 1985, as quoted in The University of Alberta's Cognitive Science Dictionary.
The point of the above quote that is relevant to us is that understanding does not come from a definition. As we begin to use the vector concept and talk about vectors in different contexts, we will develop a deeper and richer understanding. Remember how hard it is to define “energy.” Our understanding of energy continually increased as we worked with the idea in more and more contexts. Similarly with “vector,” even though we can give a short one-sentence definition of “vector,” understanding comes as we use the concept and work with it over the remainder of this and the next two chapters.
One way to represent vectors is with arrows pointing in the direction of the vector with the length of the arrow representing the magnitude. We refer to the arrowhead as the head of the vector, and the other end as the tail.
In print, in order to show the two or three dimensional vector nature of a force (or any other quantity with vector properties,) a bold symbol (such as F, r, v, a) is used or a small arrow is put over the symbol. The magnitude of a vector that could point in any direction in two or three dimensional space is normally printed in plain (not bold) italic type. Thus, the symbol “F” incorporates the vector properties (magnitude and direction), but the symbol “F” means the magnitude of the force only. Note: When a vector can point in only one direction, left or right, for example, a lower case symbol is typically used, and in an algebraic equation involving only one spatial variable (x or y for example), the symbol can take on both positive and negative values, consistent with the convention used for the spatial variable.
Vector properties usually do not include the location of the arrow, but only its length and the direction it points. Thus, the arrows, representing vectors, can be slid anywhere around on a drawing as long as the length and direction are preserved. (Although where a vector is located in space is not significant for the vector properties we are about to discuss, for some of the constructs represented by a vector, such as force, where the forces are located or where they act is very important. In these cases, we need to specify where the construct being represented by an arrow is physically located. We will typically do this using a convenient diagram.)
In the examples in figure 6.1.1, vector J is about three units in magnitude and points in a direction 45° below the +x axis, or we could say points South-East. Note that one unit length is the distance across two squares on the grid. Vector K is four units in magnitude and points in the –y direction, or South. Vector L is about two units in magnitude and points in a direction given by the angle \(\theta \) with respect to the +x axis.
The figure below shows three vectors, A, B, and C. We add two vectors by sliding one or both around so that the tail of the second is at the head of the first (it doesn't matter which one is which). The vector sum, A + B (or equivalently, B + A) is the arrow that connects the tail of the first to the head of the second. The resulting vector that represents the vector sum A + B is usually represented by a double-line arrow. More than two vectors are added by continuing to add the tail of the next vector to the head of the last one added. It doesn't matter in what order we add these vectors together, or where the location of the vector-sum arrow is. In the figure B is added to A and then C is added, but they could have been added in any order.
Subtraction of vectors is accomplished by using the relation that the negative of a vector is a vector of the same magnitude, but pointing in the opposite direction. That is, A – B is obtained by adding –B to A or A to –B, as shown in the figure). That is, A – B = A + (–B) = –B + A. This is shown in the rightmost construction in figure 6.1.2.
The basic idea of vector addition, i.e., that vector A is equal (or equivalent) to the sum of two other vectors, say B and C, is the basis of the concept of vector components. If B and C are perpendicular to each other, then we say that B is the component of A in the direction of B and that C is the component of A in the direction of C. The figure 6.1.3 shows one set of components of vector A: B and C and in different directions, another set of components, F and G.
The important thing is that the components are perpendicular to each other. We might associate the directions of the components with an x-y coordinate system. In the figure, we could orient an x-y coordinate system with the positive x direction pointing horizontally to the right. Then, B is the component of A in the x direction and points in the negative direction. Likewise, C is the component in the y direction and also points in the negative direction. The components F and G lie along the perpendicular axes of a tilted coordinate system.
The reason vector components are useful is because we are working in Galilean space-time. Each of the three components is independent of the others. Often we are concerned only with the motion of some object in a particular direction; i.e., the component of the motion in a particular direction. We can focus on the motion in this direction (this component of the motion) without worrying about what is happening in the other two perpendicular directions, since the motions are independent of each other.
Due to the independence of any three perpendicular directions in space, as well as the independence of what happens in each direction, we should think of vector quantities as really being the combination of three independent “things.” The component representation we are about to develop emphasizes the “threeness” of vector quantities. The “vector” notation—an arrow or a bold symbol—emphasizes their “oneness.” Both representations are very useful. We will frequently use a rectangular coordinate system and the associated vector components when we need to find numerical values involving vectors. Note: Frequently, we will be dealing with vector quantities in only one or two dimensions, rather than in all three spatial dimensions. That is, we are restricting the vectors in the particular physical situation to lie in only one or two dimensions, but they still represent physical quantities that exist in three dimensions.
The components of a vector are the projections of the vector onto some arbitrarily chosen set of perpendicular coordinate axes. In other words, in two dimensions, for example, we visualize the vector sitting in a 2-D coordinate system as shown, with its "tail" at the origin. Perpendiculars are then drawn from the "head" to each of the coordinate axes. These mark off the lengths Ax and Ay which are the coordinates of the vector A. We can also relate the magnitude of the vector to its components through appropriate trig functions. Ay = Asin \(\theta \) ; Ax = Acos \(\theta\). As shown in the figure 6.1.4, vector A is the vector sum of the two perpendicular components, Ax and Ay. The components and the magnitude of the vector also satisfy the Pythagorean Theorem, of course: Ax2 + Ay2 = A2.