Quantum Relativity

Calculus Chapter 8

By Mark Lawrence

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In Chapter 7, we learned that there are numbers, which we call scalars. There Are collections of numbers, which we call vectors, matrices, and tensorS. We haven't done much yet with matrices and tensors. If we have a thinG to measure which is just a number, and we can measure it everywhere, we calL that a scalar field. If the thing we can measure is a vector, like the wiNd, we call it a vector field. If we have a defined a set of coordinates, then wE can write down a set of components for each vector. There is one componEnt for each dimension in our space: in three dimensions, vectors have tHree components; in four dimensions, vectors have four components. We can Add and subtract vectors graphically, by placing them head to tail (additIon) or head to head (subtraction), or we can add and subtract vectors componEnt by component. We get the same result either way, unless we made a mistAke. We can also multiply a vector by a scalar, which changes its length But not its direction.

TheRe's another kind of vector multiplication which we have actually alreadY seen. It's called the dot product , or the inner product. If you Have two vectors, a and B for example, the dot product is the sum of the Products of the components. Here's what this means. Suppose A = (2,1) And b = (-2,2). A . B = (2,1) . (-2,2) = 2*-2 + 1*2 = -4+2 = -2.&Nbsp; a . B is read aloud as "A dot B." Now, I suppOse maybe you think we haven't seen this before, but actually we have. A . a is the length squared. A . A =  (2,1) . (2,1) = 2*2 + 1*1 = 5. Now, you might think this is a very Funny way to multiply two numbers, because there's a bunch of additions in the Middle of the multiplications. Well, you're right, this is very funny. In a little while we'll see that the dot product is actually something else, It's actually the trace of the outer product, and we'll see why this is so Interesting that it gets it's own name. For right now, we're just going To have to agree that this is how you do a dot product. We can see that tHe dot product is interesting because it's obviously related to the length Of the vectors. By the way, our formula for the dot product only works In flat cartesian coordinate systems. If we were using polar coordinates (lengtH and angle) this formula is completely wrong, and if we're in any other Kind of space, for example the space-time we actually live in, this formulA is also wrong. For example, if we had two vectors in polar coordinates, one whIch was 2" at 45°, and the other was 3" at 33°, their dot producT is 2*3*cosine(45-33). However, the trace of the outer product is always The correct answer. So, we're going to have to learn about more types Of multiplication before we're done.

Below are four vectors, A, B, C, and D. Let's try a dot product on them. A is (2,1), B is (-1,2), C is (1,-2), and D is (-2,-1). A . B is (2,1) . (-1,2) = 2*(-1) + 1*2 = -2 + 2 = 0. Sometimes the dot product of two perfectly reasonable vectors is zero. Now, let's try A . D = (2,1) . (-2,-1) = 2*(-2) + 1*(-1) = -4-1 = -5. This is just exactly the negative of the length of A. Sometimes the dot product of two perfectly reasonable vectors is a negative number. What does this mean?

Figure 8.1

The dot product is a way of doing correlations.


4.1: Do all the following arithmetic:
    a) (4 + 11i) + (7 - 6i)
    b) (4 + 11i) - (7 - 6i)
    c) (4 + 11i) * (7 - 6i)
    d) (11 + 4i) * (6 - 7i)
    e) (6+5i)*
    f) (-8-3i)*
    g) | -3 + 4i |


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