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Let F be a field. We use
F superscript n to denote the set of all n-tuples with entries in F.
For example, if you look at R^2
it is set of elements of the form a b
where a and b are real numbers.
Now the set R^2 is usually depicted by a 2-dimensional plane with two
perpendicular axes.
The horizontal axis, usually labelled as the x_1 axis,
represents the values of first entries of tuples.
And the vertical axis, usually labeled as the x_2 axis,
represents the values of second entries of tuples.
Suppose that I want to plot the tuple
-1, 2. First of all, you mark off
the values on the x-axis. So these are 1
2 and 3. And this is -1,
-2, -3. And this is 1,
2, and 3 here. And this is
-1, -2 and so on. So both axes
extend indefinitely in both directions. And what you do is,
you look at your first component of the tuple that you want to plot
and put at vertical line through the x-axis
at the value corresponding to the first entry. So in this case it's -1.
So I put a vertical line through the x_1 axis
at the value -1. And then
you look at the second entry and put a horizontal line through the x_2 axis
at the value of the second entry. So in this case it's
2. So I put a horizontal line through the x_2 axis
at the value 2. And the intersection
represents the tuple that you're plotting. So let me use green here.
Now there's a special point on this plane.
It's the intersection of the two axes and of course it represents the tuple
0,0 and this point is called the
origin. We're now going to look at what is called
the dot product of two tuples in R^n.
So take two elements from
R^n where n is a natural number.
We define u dot v
to be this quantity. And in summation notation,
this can be written as this.
So for example,
if u is 1, 2, 3,
and v is 4, 5, 6,
then u dot v
will be 1 times 4
plus 2 times 5 and finally
plus 3 times 6 and that's
4 plus 10 plus 18
and this gives us 32.
Now the dot product has one nice application and it has to do with
2-dimensional geometry. So let's go back to
our plane representing R^2 and say you have
u, v plotted on this plane.
Now, say you form a line segment from origin between each of these points
and these two line segments of course form an angle
and we always take that angle that's smaller
and let's call angle here theta.
Then, what is known is cosine
of theta is given by u dot v
divided by the square root of
u dot u times v dot v.
So for example, take these
two 2-tuples. And if you
plot these two, you have u
over here and v over here and you can see that the angle is going to be
pi/2 radians. So we expect the cosine of the angle to be 0.
And let's see what happens if we use this formula.
So if I take u dot v,
I get 2 times 0 plu
0 times -1 in the numerator.
And the denominator doesn't really matter because
the numerator is going to be 0 and so this is 0.
So that checks out. Now what if I take some other
pair u, v. Say 3, 3
and 0, 2. So this time
u is around here and
v is around there. And so
the angle, as you can see, is going to be
pi/4 and the cosine of pi/4 is 1 over the square root of 2.
So let's see if that's the case given by this formula here. So we'll be quick.
u dot v is just 0 plus 6. So it's 6.
u dot u is 9 plus 9 so that's 18.
v dot v is 4. And you now pull out the perfect square under this square root.
So 18 is 9 times 2
and 4 is a perfect square. So this will give me 6 divided by
6 times the square root of 2
and that is just 1 over the square root of 2
as expected.
Finally, if you look at the quantity given by the square root of u dot u,
it actually gives you the length of the line segment
between the origin and u.
And you can easily prove this using the Pythagorean theorem, for example.