If you're seeing this message, it means we're having trouble loading external resources on our website.

If you're behind a web filter, please make sure that the domains ***.kastatic.org** and ***.kasandbox.org** are unblocked.

Main content

Current time:0:00Total duration:10:53

Let's see if we can get a little
bit more practice and intuition of what cross products
are all about. So in the last example,
we took a cross b. Let's see what happens when
we take b cross a. So let me erase some of this. I don't want to erase all of it
because it might be useful to give us some intuition
to compare. I'm going to keep that. Actually, I can erase
this, I think. So the things I have drawn
here, this was a cross b. Let me cordon it off so you
don't get confused. So that was me using the right
hand rule when I tried to do a cross b, and then we saw that
the magnitude of this was 25, and n, the direction,
pointed downwards. Or when I drew it here, it would
point into the page. So let's see what happens with
b cross a, so I'm just switching the order. b cross a. Well, the magnitude is going to
be the same thing, right? Because I'm still going to take
the magnitude of b times the magnitude of a times the
sine of the angle between them, which was pi over 6
radians and then times some unit vector n. But this is going
to be the same. When I multiply scalar
quantities, it doesn't matter what order I multiply
them in, right? So this is still going to be
25, whatever my units might have been, times
some vector n. And we still know that that
vector n has to be perpendicular to both a and b,
and now we have to figure out, well, is it, in being
perpendicular, it can either kind of point into the page here
or it could pop out of the page, or point
out of the page. So which one is it? And then we take our right hand
out, and we try it again. So what we do is we take
our right hand. I'm actually using my right hand
right now, although you can't see it, just to make sure
I draw the right thing. So in this example, if I take
my right hand, I take the index finger in the
direction of b. I take my middle finger in the
direction of a, so my middle figure is going to look
something like that, right? And then I have two leftover
fingers there. Then the thumb goes in the
direction of the cross product, right? Because your thumb has a right
angle right there. That's the right angle
of your thumb. So in this example, that's the
direction of a, this is the direction of b, and we're
doing b cross a. That's why b gets your
index finger. The index finger gets the first
term, your middle finger gets the second term, and the
thumb gets the direction of the cross product. So in this example, the
direction of the cross product is upwards. Or when we're drawing it in two
dimensions right here, the cross product would actually
pop out of the page for b cross a. So I'll draw it over. It would be the circle
with the dot. Or if I were to draw it
analogous to this, so this right here, that
was a cross b. And then b cross a is the exact
same magnitude, but it goes in the other direction. That's b cross a. It just flips in the
opposite direction. And that's why you have to use
your right hand, because you might know that, oh, something's
going to pop in or out of the page, et cetera, et
cetera, but you need to know your right hand to know
whether it goes in or out of the page. Anyway, let's see if we can
get a little bit more intuition of what this is all
about because this is all about intuition. And frankly, I'll tell you, the
cross product comes into use in a lot of concepts that
frankly we don't have a lot of real-life intuition, with
electrons flying through a magnetic field or magnetic
fields through a coil. A lot of things in our everyday
life experience, maybe if we were metal filings
living in a magnetic field-- well, we do live in
a magnetic field. In a strong magnetic field,
maybe we would get an intuition, but it's hard to have
as deep of an intuition as we do for, say, falling
objects, or friction, or forces, or fluid dynamics even,
because we've all played with water. But anyway, let's get a little
bit more intuition. And let's think about why is
there that sine of theta? Why not just multiply the
magnitudes times each other and use the right hand rule and
figure out a direction? What is that sine of
theta all about? I think I need to clear this up
a little bit just so this could be useful. So why is that sine
of theta there? Let me redraw some vectors. I'll draw them a
little fatter. So let's say that's a,
that's a, this is b. b doesn't always have
to be longer than a. So this is a and this is b. Now, we can think of
it a little bit. We could say, well, this is the
same thing as a sine theta times b, or we could say this
is b sine theta times a. I hope I'm not confusing-- all
I'm saying is you could interpret this as--
because these are just magnitudes, right? So it doesn't matter what order
you multiply them in. You could say this is a sine
theta times the magnitude of b, all of that in the direction
of the normal vector, or you could put the
sine theta the other way. But let's think about what
this would mean. a sine theta, if
this is theta. What is a sine theta? Sine is opposite over
hypotenuse, right? So opposite over hypotenuse. So this would be the
magnitude of a. Let me draw something. Let me draw a line here and
make it a real line. Let me draw a line there,
so I have a right angle. So what's a sine theta? This is the opposite side. So a sine theta is a, and sine
of theta is opposite over hypotenuse. The hypotenuse is the magnitude
of a, right? So sine of theta is equal to
this side, which I call o for opposite, over the
magnitude of a. So it's opposite over
the magnitude of a. So this term a sine theta is
actually just the magnitude of this line right here. Another way you could--
let me redraw it. It doesn't matter where the
vectors start from. All you care about is this
magnitude and direction, so you could shift vectors
around. So this vector right here, and
you could call it this opposite vector, that's the
same thing as this vector. That's the same thing as this. I just shifted it away. And so another way to think
about it is, it is the component of vector a, right? We're used to taking a vector
and splitting it up into x- and y-components, but now we're
taking a vector a, and we're splitting it up into--
you can think of it as a component that's parallel to
vector b and a component that is perpendicular to vector b. So a sine theta is the magnitude
of the component of vector a that is perpendicular
to b. So when you're taking the cross
product of two numbers, you're saying, well, I don't
care about the entire magnitude of vector a in this
example, I care about the magnitude of vector a that is
perpendicular to vector b, and those are the two numbers that
I want to multiply and then give it that direction
as specified by the right hand rule. And I'll show you some
applications. This is especially important--
well, we'll use it in torque and we'll also use it in
magnetic fields, but it's important in both of those
applications to figure out the components of the vector that
are perpendicular to either a force or a radius in question. So that's why this cross product
has the sine theta because we're taking-- so in
this, if you view it as magnitude of a sine theta
times b, this is kind of saying this is the magnitude
of the component of a perpendicular to b, or
you could interpret it the other way. You could interpret it as a
times b sine theta, right? Put a parentheses here. And then you could view
it the other way. You could say, well, b sine
theta is the component of b that is perpendicular to a. Let me draw that, just to
hit the point home. So that's my a, that's my b. This is a, this is b. So b has some component of it
that is perpendicular to a, and that is going to look
something like-- well, I've run out of space. Let me draw it here. If that's a, that's b, the
component of b that is perpendicular to a is going
to look like this. It's going to be perpendicular
to a, and it's going to go that far, right? And then you could go back to
SOH CAH TOA and you could prove to yourself that the
magnitude of this vector is b sine theta. So that is where the sine
theta comes from. It makes sure that we're not
just multiplying the vectors. It makes sure we're multiplying
the components of the vectors that are
perpendicular to each other to get a third vector that is
perpendicular to both of them. And then the people who invented
the cross product said, well, it's still ambiguous
because it doesn't tell us-- there's always two
vectors that are perpendicular to these two. One goes in, one goes out. They're in opposite
directions. And that's where the right
hand rule comes in. They'll say, OK, well, we're
just going to say a convention that you use your right hand,
point it like a gun, make all your fingers perpendicular,
and then you know what direction that vector
points in. Anyway, hopefully, you're
not confused. Now I want you to watch
the next video. This is actually going to be
some physics on electricity, magnetism and torque, and
that's essentially the applications of the cross
product, and it'll give you a little bit more intuition
of how to use it. See you soon.