What is a number? This is a question where we think the answer is so obvious that we never even ask it. But, it turns out that not all numbers were created equal.
Suppose I asked you what the temperature was. You could look at a thermometer, which you just happen to have in your hand, and tell me. Now, suppose I asked you what the temperature was everywhere in California. We'll neglect how long it would take you to walk around everywhere - we'll suppose you can be everywhere all at once, just like Santa. You could make take a map of California and write down the temperature everywhere. We have a name for this sort of thing: if you have a thing like temperature which is defined everywhere, we call that a field. In this case, it looks like temperature is just a number, so this field consists of a single number everywhere. A single number is called a scalar. A field which is a single number defined everywhere is called a scalar field.
Now, suppose I asked you to tell me what the wind was. You can imagine you have some sort of gadget, you hold it up and it tells you the wind speed. Perhaps you would answer me by saying, "11 mph." However, I would not be happy with this answer - I'd ask you to tell me the direction of the wind, too. So, maybe you'd say "11 mph coming from 20 degrees west of north." Temperature is a single number. Wind is a single number, for the strength or speed, and also something else which tells you about the direction. Something which has a size and a direction is called a vector . Similarly, I could ask you the wind speed and direction all over California, and you could take a map and write down the speed and direction everywhere. A vector which is defined everywhere is called a vector field. There's something else to notice here. If you lived somewhere where the wind always came from the same direction, you might mistakenly think the wind was a scalar. How can you tell if something is a scalar or a vector? This can be tricky. In fact, the history of physics is littered with theories that made a mistake just like this, and had to be thrown out when the truth was finally recognized.
Finally, I could ask you to tell me something about the gravity where you are standing. You might say something like, "when I drop a small rock, it accelerates at 32 feet per second per second." But, again I would not find that satisfying (imagine living with me. . .) I would note to you that if you held up two rocks near to each other but not touching, they did not fall exactly the same way. Two rocks held out at arms length from each other, and released together, come closer to each other as they fall. Two rocks held with one five feet higher than the other get farther apart as they fall. See Figure 7.1 below. On the left is what these four rocks look like if you're standing the earth watching them fall. On the right is the same four rocks, but how it looks if you're falling with them. When you're falling with them, you don't really notice that you're falling, just that the rocks to your sides are getting closer to you, and the rocks above and below you are getting farther apart.
Figure 7.1: Rocks fall towards the center of the earth.
Why? Well, the two rocks held out at arms length are both falling towards the center of the earth, but those two directions are not parallel - the two directions meet at the center of the Earth. Two rocks dropped with five feet height difference both fall towards the center of the Earth, but the lower rock is closer and feels a little more gravity than the higher rock, so it falls just a little faster. So, we see that gravity is not like the wind - it has a strength, 32 meters per second per second, it has a direction, which is towards the center of the earth, and it has this other property, which is that as things fall, they get squeezed from the sides and stretched from top to bottom. The squeezing and stretching have a name, this is called the tidal force. It's the tidal force on the Earth caused by the Moon and Sun that cause the tides in the ocean. Tides are normally about six feet, that is the water is normally about six feet higher at high tide than at low tide, but in the Bay of Fundy between Nova Scotia and New Brunswick, tides are normally in the 40 to 50 foot range, and come in faster than a man can run in some places. So, this tide stuff is not to be ignored. Especially in Fundy, where the water is very cold. Well, anyway, we see that wind is more complicated than temperature, so temperature fits in a scalar, but wind requires a new type of object, a new type of number, called a vector. Gravity is even more complicated than wind. A size and a direction are not enough to describe gravity, you also need extra information about this squeezing and stretching stuff. Gravity fits into an object called a tensor, which is yet another type of number. Interestingly, Einstein spent a little more than a year trying to invent an early version of General Relativity which he couldn't make work, because he thought gravity was a scalar. Once he realized that it was a tensor, things progressed much more quickly. A tensor which is defined everywhere is called a tensor field - for example, we can measure gravity everywhere in California, and this would be a tensor field.
It's important that you get this point, so in homage to Strunk and White, I shall repeat it three times. Scalars, vectors, and tensors are all just different types of numbers. Scalars, vectors, and tensors are all just different types of numbers. Scalars, vectors, and tensors are all just different types of numbers.
If I'm standing in the middle of a field and I ask you, "What's the temperature where I'm standing?" you can look at your map and tell me a number, a scalar, like "85 degrees." If I turn to face the north, or the south, or any direction at all, the temperature is still 85 degrees where I'm standing. However, if I ask you about the wind, and you say "11 mph coming from your left," if I turn and face sideways, you have to adjust what you tell me. If I had turned to my left 90 degrees, you would say, "Oh, now the wind is 11 mph coming from straight ahead of you." So, we see that these vector objects have an extra piece of complication compared to a scalar: the vector depends on how you rotate, but the scalar does not. Tensors also depend on how you rotate, but in a slightly more complicated fashion than vectors do. Now we're ready to give a slightly more formal definition of vectors and tensors: a vector is any object that transforms like a vector when you rotate. A tensor is any object that transforms like a tensor when you rotate. Perhaps you once studied just a little bit of logic or philosophy. If so, you might say those definitions are circular, that is that I'm defining these objects in terms of themselves. Well, guess what: too bad. This is how physicists define them. If you don't find this satisfying, well, now you're beginning to see how I feel much of the time. Maybe by the end of this book, you'll be hard to live with too.
We know how to write a number. We write "85 degrees" by just writing "85." But, we can't write the wind velocity as "11." We need more stuff. We need to somehow describe the direction. We sometimes write down a vector by drawing a little line with an arrow at one end. The direction of the arrow indicated the direction of the vector, and the length of the arrow indicated the size of the vector. In figure 7.1, I drew little vectors that told you the direction in which the rocks were moving. So, for the wind, we'd use a little arrow pointing in the same direction as the wind is blowing, and if we had previously agreed that one inch equals ten miles per hour, the little arrow would be 1.1 inches long. This is how we draw vectors. No one has ever invented a way to draw tensors that has caught on, so there won't be much in the way of pictures for these tensor objects.
Vectors are little arrows - this is actually an important concept. We can define a vector without any reference to any coordinate system. If I ask you what the wind vector is, you can point in the direction the wind is blowing, and tell me how fast it's going. Or, if we had previously agreed that 1 inch equals 10 mph, you could make a little arrow which was 1.1 inches long, and point it in the direction the wind is blowing. So vectors have an existence which is independent of any particular observer, and independent of any particular coordinate system. This is obviously true, because the wind is simply blowing the direction it is blowing - the wind doesn't know anything about us, or how we're facing, or whether we think in miles or kilometers or furlongs. However, to work with vectors, we usually like to have a coordinate system.
Now we have a new complication - maybe you have some particular coordinate system, but I have a different one. For example, if we are standing facing each other, maybe you would say the wind was 11 mph coming from directly behind you, but I would say it was coming from directly in front of me. This disagreement between us is because we are facing in different directions, so our coordinate systems are rotated compared to each other. So, in order to talk about vectors, we're going to have to learn about coordinate systems and how to transform from one coordinate system to another. We're going to work in two dimensions for a little while, because two dimensions is easy to draw. Later, we'll work in three, then four, then lots of dimensions.
Below, in figure 7.2, is a vector in some particular coordinate system. In this coordinate system, this vector begins at the origin, and ends at the point x=2, y=1. We say this vector has components (2,1). Now, it's important to remember that the vector is the arrow, not the coordinates, not the components. The particular components we get depends on our choice of coordinate systems, but the arrow is just the arrow. So, maybe some other person is facing some different direction, he would think the vector had different coordinates, but it's still the same vector.
So, if the coordinates don't really mean that much, what exactly does mean something? We would like to know something about this vector that everyone agrees on. Well, here's some things we all agree on: there is a vector. It's pointing in the direction it is pointing. And, it has the length it has. The length is an invariant - everyone will agree on the length, no matter what their coordinate system is. What is the length? In this simple example, we know the length from Pythagoris' theorem: the length is the square root of x2 + y2. Now, we have to be careful, we remember from the complex numbers that length is not always this simple formula. This formula works only in what are called Cartesian coordinates in flat space. Cartesian coordinates are like a graph, with axis typically labeled X, Y, Z. Flat space is a mathematical presumption which we're not yet ready to discuss, but a simple definition is that if the space is flat, Pythagoris' theorem works. If we were using polar coordinates, r and θ, we would still be in flat space, but the length formula is not length2 = r2 + θ2. If space is curved, all bets are off right now. By the way, in our actual physical universe, there is no place where space is flat, so there is no place where Pythagoris' theorem actually really works. But, his theorem is pretty close pretty much everywhere, so we'll learn to use it anyway. We will remember, however, that this metric function thing is different for different people with different coordinate systems in different types of spaces. We'll also remember that the math we're learning right now doesn't work very well if we're close to a black hole.
The length of our vector above is √(4+1) = √5 = 2.2. The length of a vector is like a wind speed: everyone agrees it's 11 miles per hour. What about the direction? Well, we've already seen that people do not necessarily agree on the direction, so we cannot consider direction to be invariant. This idea, that something is invariant, this is a key idea. We'll spend some time deciding what is and what is not invariant. Much of physics is based on figuring out what's invariant. The concept of invariant does not include scale changes. So, if the wind speed is 11 mph, some other person might call it 16 feet per second, or 5 meters per second, or 18 kilometers per hour. We won't spend any time on this - in our world, everyone has agreed on one set of units.
What can we do with vectors? Well, we can do normal math - we can add, subtract, and multiply vectors. With vectors, there are several ways to multiply, which all give very different answers. We'll learn all of these ways, of course. Below are four vectors we can use for practice, called A, B, C, and D. Don't worry, they're very friendly vectors.
As suggested by the graph, we'll work in a particular coordinate system. In this system, A is (2,1), that is, A extends from the origin to the point (X=2, Y=1). The origin is (0,0). B is (-2,2), C is (-2,-2), and D is (2,-2). All of these vectors could also be thought of as being 3 dimensional vectors, with Z = 0, so we could think of A as (2,1,0), and B as (-2,2,0).
To add two vectors, you add their components. So, A+B = (2,1) + (-2,2) = (2-2,1+2) = (0,3). A+B points straight up the Y axis, and has a length of 3. B+C = (-2,2) + (-2,-2) = (-2-2,2-2) = (-4,0). B+C points straight to the right with a length of 4. Sometimes two vectors add up to zero: B+D = (-2,2) + (2,-2) = (-2+2,2-2) = (0,0). This vector is called the zero vector, or the null vector, or just 0.
To subtract two vectors, you subtract their components. A-B = (2,1) - (-2,2) = (2+2,1-2) = (4,-1). B-A = (-2,2) - (2,1) = (-2-2,2-1) = (-4,1). Notice that B-A = -(A-B). This is a good thing, it means our intuition about adding and subtracting works with vectors. B-C = (-2,2) - (-2,-2) = (-2+2,2+2) = (0,4). Now, we can guess (correctly) that C-B = (0,-4).
Remember, vectors have an existence independent of any coordinate system. When we learned how to add and subtract vectors above, we did our work in a particular coordinate system. This is not necessary, however. You can add and subtract vectors graphically, without any coordinates at all. In Figure 7.4 below, I've drawn A and B again. To add A and B, you make a copy of B and attach it to A, nose to tail. The sum of the two vectors is where the copy of B winds up - at (0,3), just as we calculated above. To subtract B from A, you use make another copy of B, but this time you put the vectors nose to nose. The difference of the two vectors is where the tail of B winds up - at (4,-1), just as we calculated above.
We've seen that to multiply a vector by -1, you multiply the components by -1. This is generally true, and one of the forms of multiplication we'll learn. To multiply a vector by a scalar, you multiply the vector's components by the scalar. So, 2*A = 2*(2,1) = (4,2). This should equal A+A if our intuition is to work, and it does. 7*A = (14,7). -2*B = -2*(-2,2) = (4,-4). Another thing we see is that if you multiply a vector by a positive number, the resulting vector points in the same direction, but has a different length. If you multiply a vector by a negative number, the direction changes: the vector points in the opposite direction, that is rotated by 180°. So, this is our first form of multiplying a vector. To multiply a vector by a scalar, you increase the length of the vector by that ratio. Alternatively, if you have a coordinate system, you multiply each component of the vector by the scalar. This kind of vector multiplication is called scalar multiplication, or scaling the vector. To use this kind of multiplication, you must have a scalar and a vector. If the vector is a one dimensional vector and has only one component, then scalar multiplication is exactly what you learned in 3rd grade.
An interesting thing to learn to do is to rotate a vector by 90°. This is very easy to do. If you have cartesian coordinates, you just exchange two coordinates, and change the sign of one of them. See figure 7.5 below. B is just A turned by 90°. A has coordinates (2,1), and B has coordinates (-1,2). So, you see I just exchanged the two numbers, and changed the sign of one of them. I could have changed the other sign, and made a vector C which was (1,-2). C is also rotated 90° from A, but clockwise, instead of counter clockwise. Finally, if I leave the components of A in their original spots, but change the signs of both from (2,1) to (-2,-1), I get D, which is the vector rotated by 180°, that is pointing in the opposite direction.
Graph paper for problem 7.1
7.2: Graphically add:
Graph paper for problem 7.2
7.3: Graphically subtract:
Graph paper for problem 7.3
7.4: Using coordinate components, find:
Check that the resulting vectors are the same as what you got graphically.
7.5: Find a vector rotated 90° from:
a) A: (3,3)
b) B: (-3,2)
c) C: (-2,3)
d) D: (3,-2)
Graph the original vectors and the rotated vectors. Make certain they're pointing in the correct direction.
Graph paper for problem 7.5