Electricity and magnetism
Cross product 1 Introduction to the cross product
Cross product 1
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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 sign 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
- 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.
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