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I've been requested to do a
video on the cross product, and its special circumstances,
because I was at the point on the physics playlist where I had
to teach magnetism anyway, so this is as good a time as any
to introduce the notion of the cross product. So what's the cross product? Well, we know about vector
addition, vector subtraction, but what happens when you
multiply vectors? And there's actually two ways to
do it: with the dot product or the cross product. And just keep in mind these
are-- well, really, every operation we've learned is
defined by human beings for some other purpose, and there's
nothing different about the cross product. I take the time to say that
here because the cross product, at least when I first
learned it, seemed a little bit unnatural. Anyway, enough talk. Let me show you what it is. So the cross product of two
vectors: Let's say I have vector a cross vector b, and the
notation is literally like the times sign that you knew
before you started taking algebra and using dots and
parentheses, so it's literally just an x. So the cross product of vectors
a and b is equal to-- and this is going to seem very
bizarre at first, but hopefully, we can get a little
bit of a visual feel of what this means. It equals the magnitude of
vector a times the magnitude of vector b times the sine of
the angle between them, the smallest angle between them. And now, this is the kicker, and
this quantity is not going to be just a scalar quantity. It's not just going
to have magnitude. It actually has direction, and
that direction we specify by the vector n, the
unit vector n. We could put a little
cap on it to show that it's a unit vector. There are a couple of things
that are special about this direction that's
specified by n. One, n is perpendicular to
both of these vectors. It is orthogonal to both of
these vectors, so we'll think about it in a second
what that implies about it just visually. And then the other thing is the
direction of this vector is defined by the right
hand rule, and we'll see that in a second. So let's try to think
about this visually. And I have to give you an
important caveat: You can only take a cross product when
we are dealing in three dimensions. A cross product really has--
maybe you could define a use for it in other dimensions or a
way to take a cross product in other dimensions, but it
really only has a use in three dimensions, and that's useful,
because we live in a three-dimensional world. So let's see. Let's take some cross
products. I think when you see it
visually, it will make a little bit more sense,
especially once you get used to the right hand rule. So let's say that
that's vector b. I don't have to draw a
straight line, but it doesn't hurt to. I don't have to draw
it neatly. OK, here we go. Let's say that that is vector
a, and we want to take the cross product of them. This is vector a. This is b. I'll probably just switch to one
color because it's hard to keep switching between them. And then the angle between
them is theta. Now, let's say the length of a
is-- I don't know, let's say magnitude of a is equal to
5, and let's say that the magnitude of b is equal to 10. It looks about double that. I'm just making up the
numbers on the fly. So what's the cross product? Well, the magnitude
part is easy. Let's say this angle is
equal to 30 degrees. 30 degrees, or if we wanted
to write it in radians, I always-- just because we grow
up in a world of degrees, I always find it easier to
visualize degrees, but we could think about it in terms
of radians as well. 30 degrees is-- let's see,
there's 3, 6-- it's pi over 6, so we could also write
pi over 6 radians. But anyway, this is a 30-degree
angle, so what will be a cross b? a cross b is going to equal
the magnitude of a for the length of this vector, so it's
going to be equal to 5 times the length of this b vector, so
times 10, times the sine of the angle between them. And, of course, you
could've taken the larger, the obtuse angle. You could have said this was the
angle between them, but I said earlier that it was the
smaller, the acute, angle between them up to 90 degrees. This is going to be sine of 30
degrees times this vector n. And it's a unit vector, so I'll
go over what direction it's actually pointing
in a second. Let's just figure out
its magnitude. So this is equal to 50, and
what's sine of 30 degrees? Sine of 30 degrees is 1/2. You could type it in your
calculator if you're not sure. So it's 5 times 10 times 1/2
times the unit vector, so that equals 25 times the
unit vector. Now, this is where it gets,
depending on your point of view, either interesting
or confusing. So what direction is this
unit vector pointing in? So what I said earlier is, it's perpendicular to both of these. So how can something be perpendicular to both of these? It seems like I can't
draw one. Well, that's because right here,
where I drew a and b, I'm operating in
two dimensions. But if I have a third dimension,
if I could go in or out of my writing pad or, from
your point of view, your screen, then I have a vector
that is perpendicular to both. So imagine of vector that's-- I
wish I could draw it-- that is literally going straight in
at this point or straight out at this point. Hopefully, you're seeing it. Let me show you the
notation for that. So if I draw a vector like this,
if I draw a circle with an x in it like that, that is a
vector that's going into the page or into the screen. And if I draw this, that is a
vector that's popping out of the screen. And where does that convention
come from? It's from an arrowhead,
because what does an arrow look like? An arrow, which is our
convention for drawing vectors, looks something like
this: The tip of an arrow is circular and it comes to a
point, so that's the tip, if you look at it head-on, if it
was popping out of the video. And what does the tail of
an arrow look like? It has fins, right? There would be one fin here
and there'd be another fin right there. And so if you took this arrow
and you were to go into the page and just see the back of
the arrow or the behind of the arrow, it would look
like that. So this is a vector that's going
into the page and this is a vector that's going
out of the page. So we know that n is
perpendicular to both a and b, and so the only way you can
get a vector that's perpendicular to both of these,
it essentially has to be perpendicular, or normal,
or orthogonal to the plane that's your computer screen. But how do we know if it's going
into the screen or how do we know if it's coming out of
the screen, this vector n? And this is where the right hand
rule-- I know this is a little bit overwhelming. We'll do a bunch of example
problems. But the right hand rule, what you do is you take
your right hand-- that's why it's called the right hand
rule-- and you take your index finger and you point it in the
direction of the first vector in your cross product,
and order matters. So let's do that. So you have to take your finger
and put it in the direction of the first arrow,
which is a, and then you have to take your middle finger and
point it in that direction of the second arrow, b. So in this case, your
hand would look something like this. I'm going to try to draw it. This is pushing the abilities
of my art skills. So that's my right hand. My thumb is going to be
coming down, right? That is my right hand
that I drew. This is my index finger, and
I'm pointing it in the direction of a. Maybe it goes a little bit more
in this direction, right? Then I put my middle finger, and
I kind of make an L with it, or you could kind of say
it almost looks like you're shooting a gun. And I point that in the
direction of b, and then whichever direction that your
thumb faces in, so in this case, your thumb is going
into the page, right? Your thumb would be going down
if you took your right hand into this configuration. So that tells us that the vector
n points into the page. So the vector n has magnitude
25, and it points into the page, so we could draw it
like that with an x. If I were to attempt to draw
it in three dimensions, it would look something
like this. Vector a. Let me see if I can give
some perspective. If this was straight down, if
that's vector n, then a could look something like that. Let me draw it in the
same color as a. a could look something like
that, and then b would look something like that. I'm trying to draw a
three-dimensional figure on two dimensions, so it might look
a little different, but I think you get the point. Here I drew a and
b on the plane. Here I have perspective
where I was able to draw n going down. But this is the definition
of a cross product. Now, I'm going to leave it
there, just because for some reason, YouTube hasn't been
letting me go over the limit as much, and I will do another
video where I do several problems, and actually, in the
process, I'm going to explain a little bit about magnetism. And we'll take the cross product
of several things, and hopefully, you'll get a little
bit better intuition. See you soon.