Electricity and magnetism
Dot vs. Cross Product Understanding the differences between the dot and cross products
Dot vs. Cross Product
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- Let's do a little compare and contrast between the dot
- product and the cross product.
- Let me just make two vectors-- just visually draw them.
- And maybe if we have time, we'll, actually figure out
- some dot and cross products with real vectors.
- Let's call the first one-- That's the angle between them.
- So let's just go over the definitions and then we'll
- work on the intuition.
- And hopefully, you have a little bit of both already.
- So what is a dot b?
- Well first of all, that's the exact same thing as b dot a.
- Order does not matter when you take the dot product because
- you end up with just a number.
- And that is equal to the magnitude of a times the
- magnitude of b times cosine of the angle between them.
- Let's look at the definition of the cross product.
- What is a cross b?
- Well first of all, that does not equal b cross a.
- It actually equals the opposite direction, or you
- could view it as the negative of b cross a.
- Because the vector that you end up with ends up flipped,
- whichever order you do it in.
- But a cross b, that is equal to the magnitude of vector a
- times the magnitude of vector b-- so far, it looks a lot
- like the dot product, but this is where the diverge is--
- times the sine of the angle between them.
- The sine of the angle between them.
- And this is where it really diverges.
- When we took the dot product, we just
- ended up with a number.
- This is just a number.
- There's no direction here.
- This is just a scalar quantity.
- But the cross product, we take the magnitude of a times the
- magnitude of b, times the sine of the angle between them, and
- that provides a magnitude, but it also has a direction.
- And that direction is provided by this normal vector.
- It's a unit vector.
- A unit vector gets that little hat on it.
- It's a unit vector, and what direction is it?
- Well, that's defined by the right hand rule.
- This is a vector.
- It's perpendicular to both a and b.
- And then you might say, a and b, the way I drew them,
- they're both sitting in the plane of this video screen, or
- your video screen.
- So in order for something to be perpendicular to both of
- these, it either has to pop out of the screen or pop into
- the screen, right?
- And when you learned about the cross product, I said there's
- two ways of showing a vector popping out of the screen.
- It looks like that because that's the tip of an arrow.
- And to show a vector going into the screen, it's like
- that because that is the back of an arrow.
- The rear end of an arrow.
- So how do you know which of these two it is?
- Because both of these vectors are perpendicular to a and b.
- That's where you take your right hand and you use the
- right hand rule.
- So you take your index finger in the direction of a, your
- middle finger in the direction of b, and then your thumb
- points in the direction of n.
- So let's do that.
- I'm looking at my hand.
- It's not an easy thing to do with your right hand, but your
- right hand is going to look something like this.
- Your index finger will go in the direction of a.
- Your middle finger goes in the direction of b.
- So that's my middle finger.
- And then my other two fingers just do what they need to do.
- I like to just bend them out of the way.
- So they just curl around my hand.
- And then what direction is my thumb in?
- My thumb-- well, actually I drew it at the wrong angle.
- My thumb is actually going in this direction, right?
- Into the page.
- This is the top of my hand.
- These are like my veins.
- Or, if I actually drew it correctly, where you would see
- your hand from side-- so it would look like this.
- You would see your pinky.
- Your palm and your pinky would be like that.
- And your other finger like this.
- Your middle finger would go in the direction of b.
- Your index finger goes in the direction of a, and you
- wouldn't even see your thumb, because your thumb is pointing
- straight down.
- But I think you get the point. a cross b, this n vector is
- pointing straight down.
- It's a unit vector.
- And this provides the magnitude.
- Unit vector just means it has a magnitude of one.
- So the magnitudes of the cross and the dot products seem
- pretty close.
- They both have the magnitude of both vectors there.
- Dot product, cosine theta.
- Cross product sine of theta.
- But then, the huge difference is that sine
- of theta has a direction.
- It is a different vector that is
- perpendicular to both of these.
- Now, let's get the intuition.
- And if you've watched the videos on the dot and the
- cross product, hopefully you have a little intuition.
- But I review it because I think it all fits together
- when you see them with each other.
- First, let's study a, b cosine of theta.
- If you watched the dot product video, cosine of theta, if you
- took, let's say, b cosine of theta.
- What is b cosine of theta?
- b cosine of theta-- and you could work it out on your own
- time-- if you say cosine is adjacent over hypotenuse, the
- magnitude of b cosine theta is actually going to be the
- magnitude of, if you dropped a perpendicular-- I'll use a
- different color here-- if you dropped a perpendicular here,
- this length right here-- that's b cosine theta.
- Let me draw it separately.
- I don't want to mess up this picture too much.
- So if that's b.
- If that's a-- And that's b.
- That's a.
- This is theta.
- b cosine theta, if you drop a line perpendicular to a, this
- is a right angle.
- b cosine theta, adjacent over hypotenuse is
- equal to cosine theta.
- So it would be the projection of b going in the same
- direction as a.
- So it would be this magnitude.
- That is b cosine theta.
- So the magnitude of that vector right there is the
- magnitude of b cosine of theta.
- So when you're taking the dot product, at least the example
- I just did, if you view it as the magnitude of a times the
- magnitude of b cosine theta, you're saying what part of b
- goes in the same direction as a?
- And whatever that magnitude is, let me just multiply that
- times the magnitude of a.
- And I have the dot product.
- Let's take the pieces that go the same direction and
- multiply them.
- So how much do they move together?
- Or do they point together?
- Or you could view it the other way.
- You could view the dot product as-- and I did this in the dot
- product video-- you could view it as a cosine of theta, b.
- Because it doesn't matter.
- These are all scalar quantities, so it doesn't
- matter what order you take the multiplication in.
- And a cosine theta is the same thing.
- It's the magnitude of the a vector that's going in the
- same direction of b.
- Or the projection of a onto b.
- So this vector right here is a cosine theta; the magnitude of
- a cosine theta.
- And they're actually the same number.
- If you take how much of b goes in the direction of a, and
- multiply that with the magnitude of a, that gives you
- the same number as how much of a goes in the direction of b,
- and then multiply the two magnitudes.
- Now, what is a, b sine theta?
- a, b, sine theta.
- Well if this vector right here is a cosine theta-- and you
- learned this when you learned how to take the
- components of vectors.
- This vector right here is the magnitude of a sine theta.
- You could rewrite this as the magnitude of a sine theta
- times the magnitude of b in that normal vector direction.
- So if you take a sine theta times b, you're saying what
- part of a doesn't go the same direction as b.
- What part of a is completely perpendicular to b-- has
- nothing to do is b.
- They share nothing in common.
- It goes in a completely different direction.
- That's a sine theta.
- And so you take the product of this with b and then you get a
- third vector.
- And it almost says, how different
- are these two vectors?
- And it points in a different direction.
- It gives you this-- sometimes it's called a pseudo vector,
- because it applies to some concepts
- that are pseudo vectors.
- But the most important of these concepts is torque, when
- we talk about the magnetic field; the force of a magnetic
- field on electric charge.
- These are all forces, or these are all physical phenomena,
- where what matters isn't the direction of the force with
- another vector, it's the direction of the force
- perpendicular to another vector.
- And so that's where the cross product comes in useful.
- Anyway, hopefully, that gave you a little intuition.
- And you could have done it the other way.
- You could have written this as b sine theta.
- And then you would have said that's the component of b that
- is perpendicular to a.
- So b sine theta actually would have been this vector.
- Or let me draw it here.
- That would make more sense.
- This would be b sine theta.
- So you could switch orders.
- You could visualize it either way.
- You could say this is the magnitude of b that is
- completely perpendicular to a, multiply the two, and use the
- right hand rule to get that normal vector.
- And we just decided that we're going to use the right hand
- rule to have a common convention.
- But people could have used the left hand rule, or they might
- have used it a different way.
- It's just a way that we have a consistent framework, so that
- when we take the cross product we all know what direction
- that normal vector is pointing in.
- In the next video I'll show you how to actually compute
- dot and cross products when you're given them in their
- component notation.
- See you in the next video.
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