Dot and Cross Product Comparison/Intuition

Linear algebra

Vector dot and cross products

Vector Dot Product and Vector Length
Proving Vector Dot Product Properties
Proof of the Cauchy-Schwarz Inequality
Vector Triangle Inequality
Defining the angle between vectors
Defining a plane in R3 with a point and normal vector
Cross Product Introduction
Proof: Relationship between cross product and sin of angle
Dot and Cross Product Comparison/Intuition
Vector Triple Product Expansion (very optional)
Normal vector from plane equation
Point distance to plane
Distance Between Planes

Dot and Cross Product Comparison/Intuition Dot and Cross Product Comparison/Intuition

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Dot and Cross Product Comparison/Intuition

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We've known for several videos now that the dot product of
two nonzero vectors, a and b, is equal to the length of
vector a times the length of vector b times the cosine of
the angle between them.
Let me draw a and b just to make it clear.
If that's my vector a and that's my vector b right
there, the angle between them is this angle.
And we defined it in this way.
And actually, if you ever want to solve-- if you have two
vectors and you want to solve for that angle, and I've never
done this before explicitly.
And I thought, well, I might as well do it right now.
You could just solve for your theta.
So it would be a dot b divided by the lengths of your two
vectors multiplied by each other is equal to
the cosine of theta.
And then to solve for theta you would have to take the
inverse cosine of both sides, or the arc cosine of both
sides, and you would get theta is equal to arc cosine of a
dot b over the magnitudes or the lengths of the products
of, or the lengths of each of those vectors, multiplied by
each other.
So if I give you two arbitrary vectors-- and the neat thing
about it is, this might be pretty straightforward.
If I just drew something in two dimensions right here, you
could just take your protractor out and measure
this angle.
But if a and b each have a hundred components, it becomes
hard to visualize the idea of an angle
between the two vectors.
But you don't need to visualize them anymore because
you just have to calculate this thing right here.
You just have to calculate this value right there.
And then go to your calculator and then type in arc cosine,
or the inverse cosine that are the equivalent functions, and
it'll give you an angle.
And that, by definition, is the angle between those two
vectors, which is a very neat concept.
And then you can start addressing issues of
perpendicularity and whatever else.
This was a bit of a tangent.
But the other outcome that I painstakingly proved to you in
the previous video was that the length of the cross
product of two vectors is equal to-- it's a very similar
expression.
It's equal to the product of the two vectors' lengths, so
the length of a times the length of b times the sine of
the angle between them.
Times the sign of the angle between them.
So it's the same angle.
So what I want to do is take these two ideas and this was a
bit of a diversion there just to kind of show you how to
solve for theta because I realized I've never done that
for you before.
But what I want to do is I want to take this expression
up here and this expression up here and see if we can develop
an intuition, at least in R3 because right now we've only
defined our cross product.
Or the cross product of two vectors is only defined in R3.
Let's take these two ideas in R3 and see if we can develop
an intuition.
And I've done a very similar video in the physics playlist
where I compare the dot product to the cross product.
Now, if I'm talking about-- let me redraw my vectors.
So the length of a-- so let me draw a.
b, I want to do it bigger than that.
So let me do it like that.
So that is my vector b.
So this is b.
That is a.
What is the length of a times the length of b times the
cosine of the angle?
So let me do that right there.
So this is the angle.
So the length of a if I were to draw these vectors is this
length right here.
This is the length of a.
It's this distance right here, the way
I've drawn this vector.
So this is, literally, the length of vector a.
And I'm doing it in R3 or maybe a version of it that I
can fit onto my little blackboard right here.
So it'll just be the length of this line right there.
And then the length of b is the length of
that line right there.
So that is the length of b.
Let me rewrite this thing up here.
Let me write it as b, the length of b times the length--
and I want to be careful.
I don't want to do the dot there because you'll think
it's a dot product.
Times a cosine of theta.
All I did is I rearranged this thing here.
It's the same thing as a dot b.
Well what is a times the cosine of theta?
Let's get out our basic trigonometry
tools-- SOH CAH TOA.
Cosine of theta is equal to adjacent over hypotenuse.
So if I drop, if I create a right triangle here, and let
me introduce some new colors just to ease the monotony.
If I drop a right triangle right here and I create a
right triangle right there, and this is theta, than what
is the cosine of theta?
It's equal to this.
Let me do it another color.
It's equal to this, the adjacent side.
It's equal to this little magenta thing.
Not all of b, just this part that goes
up to my right angle.
That's my adjacent.
I want to do it a little bit bigger.
It's equal to the adjacent side over the hypotenuse.
So let me write this down.
So cosine of theta is equal to this little adjacent side.
I'm just going to write it like that.
Is equal to this adjacent side over the hypotenuse.
But what is the hypotenuse?
It is the length of vector a.
It's this.
That's my hypotenuse right there.
So my hypotenuse is the length of vector a.
And so if I multiply both sides by the length of vector
a I get the length of vector a times the cosine of theta is
equal to the adjacent side.
I'll do that in magenta.
So this expression right here, which was just a dot b can be
rewritten as-- I just told you that the length of vector a
times cosine of theta is equal to this little magenta
adjacent side.
So this is equal to the adjacent side.
So you can view a dot b as being equal to the length of
vector b-- that length-- times that adjacent side.
And you're saying, Sal, what does that do for me?
Well what it tells you is you're multiplying
essentially, the length of vector b times the amount of
vector a that's going in the same direction as vector b.
You can kind of view this as the shadow of vector a.
And I'll talk about projections in the future.
And I'll more formally define them, but if the word
projection helps you, just think of that word.
If you have a light that shines down from above, this
adjacent side is kind of like the shadow of a onto vector b.
And you can imagine, if these two vectors-- if our two
vectors looked more like this, if they were really going in
the same direction.
Let's say that's vector a and that's vector b, then the
adjacent side that I care about is going to-- they're
going to have a lot more in common.
The part of a that is going in the same direction of b will
be a lot larger.
So this will have a larger dot product.
Because the dot product is essentially saying, how much
of those vectors are going in the same direction?
But it's just a number, so it will just be this adjacent
side times the length of b.
And what if I had vectors that are pretty perpendicular to
each other?
So what if I had two vectors that were like this?
What if my vector a looked like that and my vector b
looked like that?
Well now the adjacent, the way I define it here, if I had to
make a right triangle like that, the
adjacent side's very small.
So you're dot product, even though a is still a reasonably
large vector, is now much smaller because a and b have
very little commonality in the same direction.
And you can do it the other way.
You could draw this down like that and you could do the
adjacent the other way, but it doesn't matter because these
a's and b's are arbitrary.
So the take away is the fact that a dot b is equal to the
lengths of each of those times the cosine of theta.
To me it says that the dot product tells me how much are
my vectors moving together?
Or the product of the part of the vectors
that are moving together.
Product of the lengths of the vectors that are moving
together or in the same direction.
You could view this adjacent side here as the part of a
that's going in the direction of b.
That's the part of a that's going in the direction of b.
So you're multiplying that times b itself.
So that's what the dot product is.
How much are two things going in the same direction.
And notice, when two things are orthogonal or when they're
perpendicular-- when a dot b is equal to 0, we say they're
perpendicular.
And that makes complete sense based on this kind of
intuition of what the dot product is doing.
Because that means that they're perfectly
perpendicular.
So that's b and that's a.
And so the adjacent part of a, if I had to draw a right
trianlge, it would come straight down.
And if I were to say the projection of a and I
haven't draw that.
Or if I put a light shining down from above and I'd say
what's the shadow of a onto b?
You'd get nothing.
You'd get 0.
This arrow has no width, even though I've drawn it to have
some width.
It has no width.
So you would have a 0 down here.
The part of a that goes in the same direction as b.
No part of this vector goes in the same
direction as this vector.
So you're going to have this 0 kind of adjacent side times b,
so you're going to get something that's 0.
So hopefully that makes a little sense.
Now let's think about the cross product.
The cross product tells us well, the length of a cross b,
I painstakingly showed, you is equal to the length of a times
the length of b times the sin of the angle between them.
So let me do the same example.
Let me draw my two vectors.
That's my vector a and this is my vector b.
And now sin-- SOH CAH TOA.
So sin of theta, let me write that.
Sin of theta-- SOH CAH TOA-- is equal to opposite over the
hypotenuse.
So if I were to draw a little right triangle here, so if I
were to draw a perpendicular right there, this is theta.
What is the sin of theta equal to in this context?
The sin of theta is equal to what?
It's equal to this side over here.
Let me call that just the opposite.
It's equal to the opposite side over the hypotenuse.
So the hypotenuse is the length of this
vector a right there.
It's the length of this vector a.
So the hypotenuse is the length over my vector a.
So if I multiply both sides of this by my length of vector a,
I get the length of vector a times the sin of theta is
equal to the opposite side.
So if we rearrange this a little bit, I can rewrite this
as equal to-- I'm just going to swap them.
I have to do the dot product as well.
This is equal to b, the length of vector b, times the length
of vector a sin of theta.
Well this thing is just the opposite side as I've defined
it right here.
So this right here is just the opposite side,
this side right there.
So when we're taking the cross product, we're essentially
multiplying the length of vector b times the part of a
that's going perpendicular to b.
This opposite side is the part of a that's going
perpendicular to b.
So they're kind of opposite ideas.
The dot product, you're multiplying the part of a
that's going in the same direction as b with b.
While when you're taking the cross product, you're
multiplying the part of a that's going in the
perpendicular direction to b with the length of b.
It's a measure, especially when you take the length of
this, it's a measure of how perpendicular
these two guys are.
And this is, it's a measure of how much do they move in the
same direction?
And let's just look at a couple of examples.
So if you take two right triangles.
So if that's a and that's-- or if you take two vectors that
are perpendicular to each other, the length of a cross b
is going to be equal to-- if we just use this formula right
there-- the length of a times the length of b.
And what's the sin of 90 degrees?
It's 1.
So in this case you kind of have maximized the length of
your cross product.
This is as high as it can go.
Because sin of theta, it's a maximum value.
Sin of theta is always less than or equal to 1.
So this is as good as you're ever going to get.
This is the highest possible value when you have perfectly
perpendicular vectors.
Now, when is-- actually, just to kind of go back to make the
same point here.
When do you get the maximum value for your cosine of-- for
your dot product?
Well, it's when your two vectors are collinear.
If my vector a looks like that and my vector b is essentially
another vector that's going in the same direction,
then theta is 0.
There's no angle between them.
And then you have a dot b is equal to the magnitude or the
length of vector a times the length of vector b times the
cosine of the angle between them.
The cosine of the angle between them, the cosine of
that angle is 0.
Or the angle is 0, so the cosine of that is 1.
So when you have two vectors that go exactly in the same
direction or they're collinear, you kind of
maximize your dot product.
You maximize your cross product when they're perfectly
perpendicular to each other.
And just to make the analogy clear, when they're
perpendicular to each other you've minimized-- or at least
the magnitude of your dot product.
You can get negative dot products, but the absolute
size of your dot product, the absolute value of your dot
product is minimized when they're
perpendicular to each other.
Similarly, if you were to take two vectors that are collinear
and they're moving in the same direction, so if that's vector
a, and then I have vector b that just is another vector
that I want to draw them on top of each other.
But I think you get the idea.
Let's say vector b is like that.
Then theta is 0.
You can't even see it.
It's been squeezed out.
I've just brought these two things on top of each other.
And then the cross product in this situation, a cross b is
equal to-- well, the length of both of these things times the
sin of theta.
Sin of 0 is 0.
So it's just 0.
So two collinear vectors, the magnitude of their cross
product is 0.
But the magnitude of their dot product, the a dot b, is going
to be maximized.
It's going to be as high as you can get.
It's going to be the length of a times the length of b.
Now the opposite scenario is right here.
When they're perpendicular to each other, the cross product
is maximized because it's measuring on how much of the
vectors-- how much of the perpendicular part of a is--
multiplying that times the length of b.
And then when you have two orthogonal vectors, your dot
product is minimized, or the absolute
value of your dot product.
So a dot b in this case, is equal to 0.
Anyway, I wanted to make all of this clear because
sometimes you kind of get into the formulas and the
definitions and you lose the intuition about what are all
of these ideas really for?
And actually, before I move on, let me just make another
kind of idea about what the cross product can be
interpreted as.
Because a cross product tends to give people more trouble.
That's my a and that's my b.
What if I wanted to figure out the area of this
parallelogram?
If I were to shift a and have that there and if I were to
shift b and draw a line parallel to b, and if I wanted
to figure out the area of this parallelogram right there, how
would I do it just using regular geometry?
Well I would drop a perpendicular right there.
This is perpendicular and this length is h for height.
Then the area of this, the area of the parallelogram is
just equal to the length of my base, which is just the length
of vector b times my height.
But what is my height?
Let me just draw a little theta there.
Let me do a green theta, it's more visible.
So theta.
So we know already that the sin of this theta is equal to
the opposite over the hypotenuse.
So it's equal to the height over the hypotenuse.
The hypotenuse is just the length of vector a.
So it's just the length of vector a.
Or we could just solve for height and we'd get the height
is equal to the length of vector a
times the sin of theta.
So I can rewrite this here.
I can replace it with that and I get the area of this
parallelogram is equal to the length of vector b times the
length of vector a sin theta.
Well this is just the length of the cross product of the
two vectors, a cross b.
This is the same thing.
I mean you can rearrange the a and the b.
So we now have another way of thinking about what the cross
product is.
The cross product of two vectors, or at least the
magnitude or the length of the cross product of two vectors--
obviously, the cross product you're going
to get a third vector.
But the length of that third vector is equal to the area of
the parallelogram that's defined or that's kind of--
that you can create from those two vectors.
Anyway, hopefully you found this a little bit intuitive
and it'll give you a little bit more of kind of a sense of
what the dot product and cross product are all about.

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