Learn about Cross product 1 Video

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Learn about Cross product 1

I have been requested to do a video on the cross-product and it was a special circumstance because I was at the point on the Physics playlist. Where I had to teach magnetism anyway, so this is as good at time any to introduce the notion of the cross-product. So what is a cross-product? We’ve know about Vector addition, Vector subtraction but what happens when you multiply Vectors and there is actually two ways to do it. With the dot product or the cross-product and just keep in mind these are - well, really every operations we’ve learn is defined by human being for some other purpose. And there is 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. Seem 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 time sign that you knew before you start taking Algebra and using dots and parenthesis. So it’s literally just an X, so the cross-product of Vectors A and B it’s equal to. And this is going to seem very bizarre at first but hopefully we can get a little bit of 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 scale of quantities. Not just going to have magnitude it actually has direction and that direction we specify by the Vector N. The unit Vector N, we can put a little cap on it to show that it’s a unit Vector. That there are couple of things that are special about this direction that is specified by N. 1n is perpendicular to both of these Vectors; it is orthogonal to both of these Vectors. So we’ll think about 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 will 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 and other dimensions, but it really only has a use in three dimensions. And that is useful because we live in a three dimensional world. So let’s see, let’s take some cross-product and I think when you see it visually it will make a little more sense. Especially once you get used to the right hand rule, so let’s say that this is in Vector B, let’s say that is Vector A and we want to take the cross-product of them. So this is Vector A, this is B I will probably just switch to one color because it is hard to keep switching between them. And then the angle between them is Theta, now let’s say the length of A = 5 and let’s say that the magnitude of B = 10, that looks about double that. I am just making up the numbers on the fly, so what is a cross-product? Well that the magnitude part is easy, let’s say this angle is equal to 30° or if we want to write in Radiance. I always just because we grow up in a world of degrees - I always find it easier to visualize degrees but we can think about in terms of Radiance as well, 30° is let’s see there is three, six. It’s π/6 so we could also write π/6 radiance. But anyway this is a 30° angle, so what will be A cross B? Is going to equal to magnitude of A, so the length of this vector so it’s going to be equal to five times the length of this B vector, so times 10. Times the sign of the angle between them and of course you could’ve taken the larger obtuse angle. So 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°, so this is going to be sine of 30° times this vector N. And we’ll just it is a unit vector so I will go over what direction is actually pointing in a second. Let’s just figure out its magnitude - so this is equal to 50 and what sine of 30°? Sine of 30° is 1/2, you could type it in at your calculator if you are not sure, so it’s 5(10)(1/2) 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 into? Pointing in, so what I said earlier it’s perpendicular to both of these. So how can something be perpendicular to both of these, it seems I can't draw one. Well that is because right here where I drew A and B, I am operating in two dimensions. But if I had a third dimension, if I can 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 a vector that’s - I wish I could draw but it is really going straight in at this point or straight out at this point. Let me show you the notations 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 is going in to the page or into the screen. And if I draw this, that is a vector that is popping out of the screen and where that convention come from? It is from an arrow head, because what an arrow look like? An arrow which is our convention for drawing vectors but an arrow looks something like this. The tip of an arrow is circular and it comes to a point so that is the tip if you look at it head on, as if it was popping out of the video. And what is the tail of an arrow look like? It has fins, maybe one fin here and we have another fin right there. And so if you took this arrow and you were to put, if 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 vector that is going into the page and this is a vector that is going out of the page. So we know that N is perpendicular to both A and B, and so the only way that you can get a vector this perpendicular to both of these is it has to go. And essentially it has to be perpendicular or normal or a thoginal to this plane, to the plane - that is your computer screen. But how do we know if it is going into the screen or how do we go if it is coming out of the screen, this vector N. And this is where the right hand rule - I know this is a little bit a -- we will do a bunch of example problems. But the right hand rule, what you do is - you take your right hand. That is why it’s called the right hand rule and you take your index finger. And you point it into the direction of the first vector in your cross-product in order matters. So let’s do that, so you have to take your finger and put in the direction of the first arrow which is A and then you have to take your middle finger. And point in the direction of the second arrow B, so in this case your hand would look something like this. I am going to try to draw it; this is going to be-- pushing the abilities of my art skills. So that is my right hand, my thumb is going to be coming down, that is my hand that I drew. This is my index finger and I am pointing in the direction of A, maybe it goes a little bit of this direction. Then I put my middle finger and I kind of make an L with it or you kind of say like I'm - it is almost look you are shooting a gun. And I point that in the direction of B and then which ever direction that your thumb faces in. So in this case your thumb is going into the page. 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 in three dimensions it would look something like this. Vector A, let me see if I can give some perspective - so let me see if this was straight down. If that is a vector N, then A could look something like that; let me draw in the same color as A. A could look something like that and then B would look something like that. I am trying to draw a three dimensional figure on two dimensions - 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 how, this is the definition of a cross-product, now I am going to leave it there 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 am going to explain a little bit about magnetism. And we will the cross-product of several things and hopefully you will get a little bit better intuition, see you soon.

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I have been requested to do a video on the cross-product and it was a special circumstance because I was at the point on the Physics playlist. Where I had to teach magnetism anyway, so this is as good at time any to introduce the notion of the cross-product. So what is a cross-product? We’ve know about Vector addition, Vector subtraction but what happens when you multiply Vectors and there is actually two ways to do it. With the dot product or the cross-product and just keep in mind these are - well, really every operations we’ve learn is defined by human being for some other purpose.

And there is 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. Seem 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 time sign that you knew before you start taking Algebra and using dots and parenthesis. So it’s literally just an X, so the cross-product of Vectors A and B it’s equal to. And this is going to seem very bizarre at first but hopefully we can get a little bit of 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 scale of quantities. Not just going to have magnitude it actually has direction and that direction we specify by the Vector N. The unit Vector N, we can put a little cap on it to show that it’s a unit Vector. That there are couple of things that are special about this direction that is specified by N.

1n is perpendicular to both of these Vectors; it is orthogonal to both of these Vectors. So we’ll think about 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 will 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 and other dimensions, but it really only has a use in three dimensions.

And that is useful because we live in a three dimensional world. So let’s see, let’s take some cross-product and I think when you see it visually it will make a little more sense. Especially once you get used to the right hand rule, so let’s say that this is in Vector B, let’s say that is Vector A and we want to take the cross-product of them. So this is Vector A, this is B I will probably just switch to one color because it is hard to keep switching between them. And then the angle between them is Theta, now let’s say the length of A = 5 and let’s say that the magnitude of B = 10, that looks about double that. I am just making up the numbers on the fly, so what is a cross-product?

Well that the magnitude part is easy, let’s say this angle is equal to 30° or if we want to write in Radiance. I always just because we grow up in a world of degrees - I always find it easier to visualize degrees but we can think about in terms of Radiance as well, 30° is let’s see there is three, six. It’s π/6 so we could also write π/6 radiance. But anyway this is a 30° angle, so what will be A cross B? Is going to equal to magnitude of A, so the length of this vector so it’s going to be equal to five times the length of this B vector, so times 10. Times the sign of the angle between them and of course you could’ve taken the larger obtuse angle. So 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°, so this is going to be sine of 30° times this vector N. And we’ll just it is a unit vector so I will go over what direction is actually pointing in a second. Let’s just figure out its magnitude - so this is equal to 50 and what sine of 30°? Sine of 30° is 1/2, you could type it in at your calculator if you are not sure, so it’s 5(10)(1/2) 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 into? Pointing in, so what I said earlier it’s perpendicular to both of these. So how can something be perpendicular to both of these, it seems I can't draw one. Well that is because right here where I drew A and B, I am operating in two dimensions. But if I had a third dimension, if I can 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 a vector that’s - I wish I could draw but it is really going straight in at this point or straight out at this point.

Let me show you the notations 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 is going in to the page or into the screen. And if I draw this, that is a vector that is popping out of the screen and where that convention come from? It is from an arrow head, because what an arrow look like? An arrow which is our convention for drawing vectors but an arrow looks something like this. The tip of an arrow is circular and it comes to a point so that is the tip if you look at it head on, as if it was popping out of the video. And what is the tail of an arrow look like? It has fins, maybe one fin here and we have another fin right there.

And so if you took this arrow and you were to put, if 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 vector that is going into the page and this is a vector that is going out of the page. So we know that N is perpendicular to both A and B, and so the only way that you can get a vector this perpendicular to both of these is it has to go. And essentially it has to be perpendicular or normal or a thoginal to this plane, to the plane - that is your computer screen. But how do we know if it is going into the screen or how do we go if it is coming out of the screen, this vector N. And this is where the right hand rule - I know this is a little bit a -- we will do a bunch of example problems.

But the right hand rule, what you do is - you take your right hand. That is why it’s called the right hand rule and you take your index finger. And you point it into the direction of the first vector in your cross-product in order matters. So let’s do that, so you have to take your finger and put in the direction of the first arrow which is A and then you have to take your middle finger. And point in the direction of the second arrow B, so in this case your hand would look something like this.

I am going to try to draw it; this is going to be-- pushing the abilities of my art skills. So that is my right hand, my thumb is going to be coming down, that is my hand that I drew. This is my index finger and I am pointing in the direction of A, maybe it goes a little bit of this direction. Then I put my middle finger and I kind of make an L with it or you kind of say like I'm - it is almost look you are shooting a gun. And I point that in the direction of B and then which ever direction that your thumb faces in. So in this case your thumb is going into the page.

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 in three dimensions it would look something like this. Vector A, let me see if I can give some perspective - so let me see if this was straight down. If that is a vector N, then A could look something like that; let me draw in the same color as A. A could look something like that and then B would look something like that. I am trying to draw a three dimensional figure on two dimensions - 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 how, this is the definition of a cross-product, now I am going to leave it there 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 am going to explain a little bit about magnetism. And we will the cross-product of several things and hopefully you will get a little bit better intuition, see you soon.

Transcription by: Scribe4you Transcription Services