Vectors and Complex Numbers
History
As mentioned in the last section, vectors are entities that represent both a magnitude and a direction. Vector-like concepts were mentioned in the ancient times between 400BC and 100BC, but no clear conceptual framework or notation was to be introduced for another 1900 years. Newton himself used vectorial entities, but did not describe vectors mathematically. Rather, he did tedious calculations using geometry and algebra.
The practice of using complex numbers to describe a location within a plane was the precursor to vectors. Several mathematicians and scientists did this in the early 1800s, including Wessel, Argand and Gauss. You probably have experience plotting complex numbers as points in a 2-dimensional plane. The idea is to use the x-axis as the real axis, and the y-axis as the imaginary one.
The most interesting historical achievement to me is that of William Rowan Hamilton who in 1843 developed a mathematical entity called the quaternion for the application to dynamics in 3D space. Due to reasons that require a deep mathematical background, it turns out that vectors as we use today (and that we will use in this course) are somewhat awkward for 3D problems. If that sounds strange, it should. But the fact remains that 3D vectors applied to 3D problems don't work as well as 4D (really one real scalar and three imaginary components) work much better.
Due to reasons that you can read about elsewhere, the only dimensions that allow for what's called an associative normed division algebra are 1, 2, and 4 (not 3). To do mathematics in which the concepts of addition, subtraction, multiplication, and division make sense, and in which magnitudes like distance have meaning, we require one of those dimensions. The fact that the world uses 3D vectors makes for awkward mathematics. Among the "problems" are the lack of the notion of division of vectors, the lack of a clear sense of multiplication of vectors (there are different ways defined), and the difficulty of mathematically rotating objects in 3D space. There are ways in which to do these things awkwardly, but the highly underappreciated achievement of Hamilton was to recognize the need of a 4D entity called a quaternion for these purposes. You can read more about them here: Quaternion - Wikipedia. On my to-do list is to write a few chapters on physics and quaternions to illustrate their power. Until I do, have a look around the web for references to read.
In any case, if we go back to complex numbers as a representation for 2D space, you can see in the plot below, the complex number 'r' is representing a point in the xy plane. You can move the point and the corresponding value of 'r' will be shown. Just left-click and drag the point around.
A point in a 2-dimensional plane describing a complex number
Complex Numbers
When we need a quantity to not only convey a magnitude, but also a direction, it will require at least two individual numbers, as is the case for the complex number above. One of the numbers is the real part of 'r' and the other number is the imaginary. Since real and imaginary can't be added, there is no way to simplify the expression for 'r'. I just mean that it's NOT legal to write or something silly like that. We have to leave it as two terms.
It would be fine to use a complex number to represent a location. Suppose, for instance, that you wish to describe your location with respect to the college campus. The complex part of 'r' could represent how far north (+) or south (-) you are from the college, and the real could represent how far east (+) or west (-) you are. Together, the two terms might look like this:
If we decide that the real axis points eastward and the complex northward, we could understand this complex number to indicate a location that is 3.0 km eastward and 4.0km southward (the negative of northward) from the college. With that information, it would be possible to find you. The total walk to your location by first walking eastward and then southward would be 3.0km+4.0km=7.0km. Note that I am not including the 'i' here when I find how far it is along that path, since the 'i' only serves to differentiate north/south from east/west.
Of course another way to get to your location would be to walk southeastward at an appropriate angle (assuming no bulidings block the route). It is instructive to ask how far we'd need to walk, and at what angle we'd need to walk. The answer is obtained from simple trigonometry.
Since at first we walked along two directions that are at right angles to one another, they may be seen as the sides of a right triangle. The direct path is the hypotenuse of the triangle. Thus the distance walked along the diagonal can be found by the Pythagorean theorem. That would come out to This is the same thing as what you learned in math class to be the magnitude of a complex number, or
Regarding the angle, we first have to decide where to call zero. In trigonometry, in the context of the unit circle, it is conventional to use the +x direction as zero and to measure positive angles counterclockwise. Thus we know that our walk would be at a negative angle (or a large, positive one). The angle is found by using the tangent function: Notice that we should leave the negative in there so that the angle your calculator gives is also negative.
Differing Notations
Complex numbers are notated many ways depending on the textbook you are reading. Sometimes complex numbers (and vectors) are written like (3,-4). This is not ideal. It is too generic. How are you to know, apart from context, whether (3,-4) represents values of x and y or a pair of scores in a game, or any other pair of quantities? You don't. It is for this reason that complex numbers (and vectors) in more advanced texts don't use that notation. On the other hand z=x+iy, as used above, is not ambiguous. Even out of context we understand what it means.
The Birth of Vectors
In principle, we could use complex numbers to describe physics in 2 dimensions, since 2-dimensional vectors have many of the same properties as complex numbers. But to do physics in general, we need a third spatial dimension, and complex numbers only describe locations in 2 dimensions.
The vectors we use today are the imaginary components (3 of them) of William Hamilton's quaternions. His quaternions have 4 components which are written as the sum of a scalar and a vector. By 'sum', I don't mean that it's ok to add a scalar to a vector, but that a quaternion is a composite number containing both a scalar and vector part in the same way that a complex number can be thought of as a 'sum' of a real and imaginary part. In both cases you can't resolve the sum to get a simpler form. In a fundamental sense we can't add scalars to vectors since they are fundamentally different quantities.
In the case of Hamilton's quaternions, the vector part was later extracted and used on its own. Using vectors made certain laws of electromagnetism and other laws of nature simpler to handle than when described in terms of quaternions, or so the common story goes. Maxwell's equations of electromagnetism which we'll study during second semester were written in unclear notation with several mistakes by Maxwell. It is said that he used quaternions which were later simplified by Heaviside as vectors, but the reality is that the vector form of Maxwell's equations are missing a lot of information as compared with the original form. Thus we have other rules and definitions besides Maxwell's equations. The other issue is that to my knowledge nobody has ever tried to fix his bad notation and write his laws as succinctly as possible (and without errors) in terms of quaternions alone. As such we could make a better assessment of pros and cons to address vectors vs quaternions for physics. But for this period of history we will be using vectors. Flawed as they may be, they are still very powerful. They also do for us what scalars cannot.
The mathematics is also not as bad as I make it sound above. We will not over-simplify anything and will have an elegant language for the description of the laws of nature. Whether a more elegant form is possible with quaternions is a question the future may answer for us.
The Vector
A vector is just like the complex number in that it has parts that can't be added for the sake of simplification, just as it doesn't make sense to add real to imaginary. In the case of a vector, however, the parts are all real. The reason they can't be added is because the real quantities are multiplied by unit vectors that describe a direction. Just as while walking it doesn't make sense to add north to east - since they are fundamentally different directions - so it does not make sense to add together different unit vectors.
The complex number we used to describe a location, could be written as a rather similar vector: The meaning is identical, but the notation is different. Note the subtleties. You must never leave off the arrow above the 'r', for it indicates the difference between a normal scalar quantity and a vector one. Note also that instead of real and imaginary, we now have real values multiplied by unit vectors (denoted with the "hats") which describe direction. Thus that vector indicates that 'r' points to a direction 3km in the direction of the x-axis and -4km in the direction of the y-axis. It is common to use , , and instead of , , and The ijk notation came from Hamilton's quaternions. In a quaternion i, j, and k are all imaginary numbers. In vectors they are not. That means the hat (^) over the symbols is absolutely necessary! Without the hat the mathematical community still uses i and j as imaginary numbers, so the meaning is different.
I will use the two notations interchangeably at times. Since we just described complex numbers, please note that while the imaginary 'i' looks rather similar to the , that they mean very different things. One denotes the square root of negative one and the other indicates a direction in space.
Vector Components
When we refer to the individual parts of a vector, we call them components. Linguistically, this is no different than talking about the parts of a bicycle or other machine. It is common to ask "what components do you have on your bike?" In the case of a vector in two dimensions there are up to two non-zero components, and as we'll see in the next section, there are up to three non-zero components for a 3D vector.
Given a vector named , for instance, its x-component is denoted . Notice that the component itself is a scalar! In the position vector describing your location with respect to campus, we could write and Notice that components can be negative, but vectors themselves can never be negative. To negate a vector just means its direction is being flipped. It still has a positive magnitude and a direction just as it did before flipping.
A Vector's Components Calculated
Vectors can be thought to "point"
A vector "points" in a direction, and it is often thought of as an arrow. In the same example of where you are with respect to the college, the idea would be that the vector is a giant arrow pointing to your location. The arrow would start at the college and have its tip at your location. The length of the vector (arrow) is its magnitude. It would be the distance from campus to your location that we calculated above, or 5.0km. The angle the vector points would also be the same angle as we calculated above with the complex number.
A vector version of the same example is shown below. The point can be moved and the vector's corresponding value is shown. Pay close attention to the notation... the hats and the arrow. They are important. Please excuse the computer for the +-. While correct, it looks funny to us and we wouldn't write the expression that way by hand.