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>> Hello, I'm Dr. Simon Romero, and I am here to talk
to you a little bit about the dot product.
The dot product is a very important operation
between vectors that I --
we are actually going to use the entire semester.
We're going to start with the idea of the dot product;
the angle-magnitude definition of the dot product;
component definition of the dot product; direction angles
and cosines; and components and projections.
As always, I suggest you do all the practice problems
so you can detect whether you are understanding these very
important concepts or not.
The idea of the dot product is relatively simple,
but it has some details that you have
to be very careful to consider.
We want to define a product of two vectors to be a number
that measures into what extent they point
to the same direction or not.
Let me give you an example.
In this case, we can think
that the two vectors have compatible directions;
in other words, they are kind of helping each other.
In this case, we want the dot product to be positive.
In the case when they are not --
they have incompatible directions,
we want to have the dot product to be negative.
In the case when they are not helping each other,
or they are not hurting each other, we are going
to want the dot product, we want the dot product to be zero.
In every case, it seems that the angle
between the vectors is a factor
that distinguishes each situation.
When it's positive, it's -- the angle seems to be acute.
When it's negative, the angle seems to be obtuse.
And when it's zero, the angle seems to be a right angle.
If you remember, we have a trigonometry function
that actually behaves more or less like this.
And that's the cosine.
When it's acute, it's positive; when it's zero, when you are
in the right angle; and when the angle is obtuse, it's negative.
So the dot product seems that the cosine is going
to play an important role in the dot product.
Let me give you the angle definition of dot products.
Before doing that, let me just tell you that remember always
that a vector dot a vector is always going
to give you a scalar.
So in other words, the dot product
of two vectors is always going to be a number.
Dot product angle-magnitude definition:
if you have two non-zero vectors,
we define the dot product as v dot w to be the magnitude of v,
the magnitude of w and times the cosine
of the angle between them.
If one of the vectors is zero, then the cosine --
I mean, the angle between them doesn't make sense to talk
about the angle, we define the v dot zero to be equal to zero.
We're not saying -- here we come to a particular definition,
we're going to say that two vectors are orthogonal,
if the angle between those two vectors is pi over 2;
the way that we're going to represent
that algebraically is going to be when the dot product of v
and w is equal to zero.
So if the angles are orthogonal, immediately we have to think
that the dot product is equal to zero.
Let me give you the component definition of the dot product.
Given two vectors and you have the components
of the two vectors, then v dot w is equal to v --
the first component of v times the first component
of w. Then we multiply the second component of v
with the second component of w; and then the third component
of v with the third component of w;
and then we add everything together
and that gives you the dot product.
So the question is, are these two ways
to computing the dot product equal?
Well, in the class activity that we're going to have next class,
we are going to see that effectively these two methods
to compute the dot product are the same.
The difference is that this component definition is way more
useful, because usually we're going to have vectors given
by components and not always is easy
to compute the angle between vectors.
So in this one, this definition is going to be way more useful.
Let me give you some properties of the dot product.
If u, v and w are vectors in R3 and c is a scalar,
this is what is going to happen.
This one, this first one is very important.
If you take the dot product between two numbers,
then the result is the magnitude square.
And you can see using either
of the definitions that that's the case.
So one thing, one very common mistake is writing this.
Please don't write this; this doesn't make sense.
This is wrong.
v dot w is equal to w dot v, so that means
that this operation is commutative; in other words,
it doesn't matter which one you take the dot product --
I mean, which order you take the dot product.
Is distributive, but is the one that we have right
after the commutativity.
This is the commutativity, this is distributive,
but it says that you can take common factors
or you can FOIL using the product.
The other one is if you take a scale version of v
and then take the dot product with w, it's the same
as scaling first w and then v, or multiplying c
by the result of the dot product.
This one we already talked a little bit about it.
And this one at the end is going to be that the largest --
that the absolute value
of the dot product can be is the magnitude
of v times the magnitude
of w. Let me give you some applications of the dot product.
One of them is finding the angle between vectors.
Let's try to do this problem.
Find the angle between the vector v and the vector w.
And now the thing that you have to know,
this is that since the components of the vector v
and w are [inaudible],
we can use the component definition of the dot product.
In other words, I am going
to compute the dot product using those components,
so we have the -- v dot w is going to be equal to 1.
Now we -- now that we have the result this dot product,
the thing that we can do is we can use the other definition,
because the other definition involves the angle.
So we can find the angle because we already have the dot product
by just solving for the angle.
So notice something here, our dot product was equal to zero,
but on the other hand we have that the dot product is equal
to the magnitude of v, magnitude of w and the cosine
of the angle between them.
So we have that the magnitude of v, magnitude of w,
cosine of the angle between is equal to 1, so the cosine has
to be equal to 1 over the multiplication
of the magnitudes v and w. Now we just compute the magnitudes
of v and w, and we obtain the angle is the cosine of 1 over 3.
If for some reason you cannot compute arc cosines, or this,
in particular this inverse cosine,
that we cannot actually compute without the calculator.
So if that's the case,
you should leave it indicated just like I did over here.
In the case that you have one of the angles
that we can compute the inverse cosine, you should do so.
Let me talk a little bit about projections.
A projection of w onto v is the vector obtained
by dropping a perpendicular line containing v from the tip
of w. Now, let me repeat that,
I am saying that [inaudible] is the vector obtained by dropping
up a perpendicular line to the line containing v from the tip
of w. I am sorry about that.
So, and graphically it's very easy.
You just drop up a perpendicular line to the line containing v,
and then kind of like the shadow vector that you paint is going
to be the projection of w onto v. For example,
in this other case, you take the line containing --
the line containing v, drop up a perpendicular starting
from the tip of w and the shadow you obtain is a projection
of w onto v.
Here are the two cases, and the thing that we're saying is --
let me just, before continuing --
let me just tell you really quick how this notation goes.
The vector that you are projecting is the one the goes
in full size.
The vector that you're projecting onto is the one
that goes in the subscript.
So note that if the angle between the vectors is acute,
the projection and v have the same direction.
Notice that this is kind of like it's going to have a relation
with the dot product, and you will see what's the relation
later on.
Now, if the angle between the vectors is obtuse,
then the projection and v have opposite directions.
Now, could you tell me -- come tell me during class
or just think about it --
what happens when you have a right angle?
What will be the projection?
Well, I'm going to move on.
The components.
The component of w to v is the signed magnitude
of the projection of w onto v. In other words,
the absolute value of the component is equal
to the magnitude of the projection.
Let me explain a little bit more what do I mean
by signed magnitude.
So this is the component.
For example, in the other case, when they have an obtuse angle,
that's going to be the component.
Really quick, when I talk about signed magnitude, what I mean is
that the angle, if the angle is acute,
the component is going to be positive.
In other words, if the projection and v go
to the same direction, they are going to --
then the component is going to be positive.
In the case that you have an obtuse angle,
the component is going to be a negative scalar.
Notice that this is just like the dot product.
So if the component is positive,
the angle between the vectors is acute.
If the component is negative,
the angle between the vectors is obtuse.
Let me give you the formulas.
The formulas are going to be like this.
I mean, you may notice here
that the dot product it plays a very important role
to determine the sign of the component.
One way that you can remember this is the component relies
on whether the v and w are helping each other.
If they are helping each other that means
that the dot product is positive;
if they are hurting each other, then they --
or they are going against each other somewhat,
then the component's going to be negative.
And the projection is just going to be this one over here.
Again, you may wonder, well,
how these formulas are the way they are.
Well, that we're going to see in the class activities,
so don't miss that one.
Once you know where these formulas come from,
they will be very easy to recall.
It's just a little bit about trigonometry.
Let me give you an example.
An example is, find the component and the projection
of the vector, of the vector 2i minus 3j plus k
onto the vector i plus 6j minus 2k.
Now, the thing that you have to, the thing that you have
to remember is that it is very important to --
if you want to use the formulas -- to check what order, I mean,
what are, who is w and who is v?
The thing that you have to remember is that the vector
that you're projecting is what we're calling here "w,"
and the vector that you're projecting
onto is what we're calling "v."
So in this case, the vector that you're projecting is this one,
so that will be w. And the vector that you're projecting
onto is going to be v. Once we have that,
we're going to apply the formulas.
The easiest formula to remember is the component.
So let's try to do that one.
So the component given that the dot product of v dot w is equal
to negative 19, and the magnitude of v is equal
to the square root 41, then we have that the component of w
onto v is negative 18 over the square root of 41.
Now, we can use the same information
to compute the corresponding projection.
So the thing that we have is the projection of w onto v is going
to be just minus 18 over -- and instead of having the magnitude
of v, you have the magnitude of v square --
so you will have 41 instead of the square root of 41.
That's one of the difference.
And the other difference is you are multiplying by the vector v.
And notice that in this case, you obtain a vector that goes
in the opposite direction of v. Remember
that you can always pause this video,
you can go back, you can review.
And please let me know if you have any questions.
Thank you very much, and have a good one.