We've so far described abstractly
what we mean by a dot product, by a definition of AB
equals the magnitude of A times cosine theta,
times the magnitude of B. But many times in physics problems,
we actually have vectors in space
and we want to see how to do this
in terms of a Cartesian or any coordinate system
in particular.
So let's set up a coordinate system, we'll call it
Cartesian, a and j.
And now this is very important.
Let's define a vector A here, and let's have another vector
in a completely different point, B. We still can take the dot
product of these two vectors.
But in order-- this is our plus x, plus y.
And I'm only doing things in two dimensions.
So how do we calculate the dot product?
Well, the first thing that we want to look at
is our unit vectors.
What are the dot products of unit vectors?
Well, if we take i hat dot i hat, that's the magnitude of i
hat times cosine of the angle 0, times the magnitude of i hat.
And cosine of 0 is 1 and the magnitude of unit vectors
are 1.
So when you dot product the unit vector with itself, you get 1.
And therefore, it's also true for the j hat. j
hat dot j hat is 1.
What happens when you dot product two vectors
that are perpendicular?
Well, in this case, this is 0 because the angle theta
is 90 degrees.
And remember that cosine of 90 degrees is 0.
And anyway, when two vectors are perpendicular,
there's no component of one vector
that's parallel to the other.
So these are the essential facts that we're
going to need to calculate the dot product of two vectors that
are separated in space.
So the way we do that is we'll begin by drawing, writing down
the vectors in Cartesian coordinates, where
Ax A is a scalar and the vector part is and the unit vector.
And we have Ay j hat.
And likewise, we can write B as Bx i hat plus By j hat.
And now when we take the dot product of these two vectors,
we're going to write out all the terms
here so that you see them.
So here is our scalar or dot product.
We use those words interchangeably.
Now notice that we've already shown
that the dot product distributes over vector addition.
And also, if you multiply a scalar by a vector,
you can pull the scalar out.
So there's four terms here-- Ax i hat dot Bx i hat plus ax
i hat dot By j hat plus Ay j hat dot Bx i hat.
This is a little tedious to write out.
By j hat.
And now because these are scalars,
we can pull them out and the only part of a dot product
that matters is how the unit vectors dot.
And that's why we have these two results.
i hat dot i hat is 1, j hat dot j hat is 1,
and i hat dot j hat is 0.
So the first turn is Ax Bx.
i hat dot j hat is 0, so we don't need that.
j hat dot i hat is 0.
And finally j hat dot j hat is 1.
And so we get plus Ay By.
And that's how we define the dot product
in Cartesian coordinates of two vectors.
Now, notice that if we dotted a vector,
A dot with itself, that would just be Ax Ax plus Ay Ay, which
is the components squared.
And that's equal to the magnitude
of the vector squared.
And so we can say that the magnitude of the vector,
of any vector, is you take its dot product with itself.
You take the square root, but remember
we always take the positive square root, because magnitudes
are positive.
And that's how we calculate the scalar product for vectors.
And many times in the application of physics,
when we have physical quantities that
are vectors in different places, we use vector decomposition
and we use this procedure.
This is the Cartesian picture, we'll
learn how to do that in polar coordinates
when we need it later on.