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Current time:0:00Total duration:19:14

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.