Electricity and magnetism
Cross Product 2 A little more intuition on the cross product.
Cross Product 2
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- 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
- 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.
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