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Current time:0:00Total duration:10:02

Video transcript

I've been requested to do a video on the cross product and it was a it's 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've 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 a vector a cross vector B and the notation is literally like the time 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 it's 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 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 and the unit vector n we could put a little cap on it so that it's a unit vector and the the there are a couple of things that are special about this direction that's specified by N 1 n is perpendicular to both of these vectors it is orthogonal to both of these vectors so we'll think about a second what that implies about it just visually and then the other thing is the the direction of this vector is defined by the right-hand rule and we'll see that in a second so so let's 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 product 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 this is let's say that's vector B I don't have to draw straight lines but it doesn't hurt to oh I'd have to draw it neatly oh that was unusually I thought I was using the line tool okay here we go let's say that that is vector a and we want to take the cross product of them so 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 that the length of a is I don't know let's say it's you know magnitude of a is equal to five and let's say that the magnitude of B is equal to ten 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 and 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 three six it's PI over six 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 so the length of this vector so it's going to be equal to five times the length of this B vector so times ten times the sine of the angle between them and of course there's you could have 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 this vector N and we'll just 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 new calculator if you're not sure so it's 5 times 10 times 1/2 times the unit vector I don't like that color 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 is it's perpendicular to both of these so how can something be perpendicular to both of these you know on 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 if I have a third dimension if I could go in or out of you know my writing pad or from from your point of view your your screen then I have a vector that is perpendicular to both so imagine a vector that's I wish I could draw it that you know is literally going straight in at this point or straight out at this point hopefully you're saying 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 from an arrowhead because what does 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 it's 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 then what does the tail of an arrow look like it has fins right it'd be one fin here and then to be another fin right there and so if you took this arrow and you were to put if you were to if you were to go into the page and just see the 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 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 is it has to go it essentially has to be perpendicular or normal or orthogonal to this plane to the plane that's your computer screen but how do we know if it's going in to into the screen or how do we go 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 a 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 in the direction of the first arrow which is a and then you have to take your middle finger and point it in the 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 see my hand let's see so your this is going to be so might let me this is pushing the 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 of this direction right then I put my middle finger and I kind of make an L with it or you could kind of say like I'm 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 end points into the page so the vector end 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 so let me see if this was straight down if that's a vector n then a could look something like that let me draw in the same colors 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 how the this is 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