Cross product introduction

we've learned a good bit about the dot product the dot product but when I first introduced it I mentioned that this is only one type of vector multiplication and the other type is the cross product which are probably familiar with from your vector calculus course or from your physics course the cross product but the cross product is actually much more limited than the dot product it's useful but it's much more limited the dot product is defined in any in any dimension so this is defined for any two vectors that are in any two vectors that are in RN you could take the dot product of vectors have two components you could take the dot product of vectors that have a million components the cross product is only only defined defined in r3 and then the other I guess major difference is the dot products we're going to see this in a second when I define the dot product for you I haven't defined it yet the dot product results in a scalar when you take the dot product of two vectors you just get a number but when the cost product you're going to see that we're going to get another vector and the vector we're going to get is actually going to be a vector that's orthogonal to the two vectors that we're taking the cross product of so now that I have you excited with anticipation let me define it for you and you probably already have seen this once or twice in your mathematical careers let's say I have the vector a and it's only let have to be an r3 so it only has three components a 1 a 2 and a 3 and I'm going to cross that with the vector B and it has three components B 1 B 2 and B 3 a cross B a cross B is defined as a third vector and now this is going to seem a little bit bizarro and hard to essentially memorize because this is a definition but I'll show you how I think about it when I have my vectors written in this column form if you watch the platelet the physics play let's have a bunch of videos on the cross product and I show you how to how I think about the cross product I have it in kind of the ijk form but when I do it when I have it like this the way you think about it this first term up here this is going to be another three vector or another vector in r3 so it's going to have one two three terms for the first term what you do is you ignore these top two terms of this vector and then you look at the bottom two and you say a2 times b3 minus a3 times b2 if you've done any of the if you've if and I've made a few videos on determinants although I haven't formally done them in kind of this linear algebra playlist yet but if you remember kind of cofactor exciting out the cofactor terms for when you're determining the determinants or if you're just taking the determinant for a two by two matrix this might seem very familiar so this first term right here is essentially the determinant of if you if you get rid of this first this first row out of both of these guys right here you just take a two times b3 minus a3 times b2 so it's a two times b3 minus a3 times b2 that was hopefully pretty straightforward now if not to make your life any more complicated when you do the second when you do the middle row when you do this one right here so you cross that out and you might want to do a 1 times b3 minus a3 times b1 and that would be natural because that's what we did up there but the middle row you do the opposite you do the you do 830 times b1 minus a 1 times b3 or you can kind of view it as the negative of what you would have done naturally so you would have done a 1 b3 minus a3 b1 now we're going to do a three b1 a3 b1 minus a1 b3 minus a1 b3 and that was only for that middle row and then for the bottom row we cross that out again or ignore it and we do a 1 times b2 just like we did with the first row a1 b2 times a2 b1 or minus a2 b1 a to be won and this seems all hard to and it is hard to remember that's why I kinda have to get that system in place like I just talked to you about but this might seem pretty bizarre and hairy so let me do a couple of examples with you just so you get the hang of our definition of the dot product in r3 so let's say that I have the vector let's say I'm crossing the vector I have the vector 1 - 7 + 1 and I'm going to cross that with the vector 5 - 4 so this is going to be equal to 1/3 vector this is going to be equal to 1/3 vector let me get some space to do my mathematics so for the first element in this vector the first component we just ignore the first components of these vectors we say - 7 times 4 minus 7 times 4 minus 1 times 2 minus 1 times 2 and these are just regular multiplication I'm not taking the dot product these are just regular numbers then for the middle term we ignore the middle terms here and then we do the opposite we do 1 times 5 1 times 5 minus 1 times 4 minus 1 times 4 remember the first you might have been tempted to do 1 times 4 minus 1 times 5 because that's how we essentially did it in the first term but the middle term is the opposite and then the finally the third term you ignore the third terms here and you do it just like the first term you start on the top left 1 times 2 1 times 2 minus minus 7 I'll put that in parentheses minus minus 7 times 5 and so that is equal to that is equal to let me see what do we get - 7 times 4 I want to make a careless mistake here that's - 28 - - right - 2 so this is minus 34 that first term this one is 5 - 4 so it's -5 - 4 so it's a sorry 5 minus 4 is just 1 and then 2 let's see minus minus 35 so 2 minus minus 35 this 2 plus 35 that's 37 so there you go hopefully you understand at least the mechanics of the cross product so the next thing you're saying well ok I can I can find the cross product of two things but what is this good for what does this do for me and the answer is is that this third vector right here depending on whether I stay in the abstract case or whether I this is the case with numbers this is orthogonal to the two vectors that we took the cross product of so this vector right here is orthogonal orthogonal to a and B right that a and B which is pretty neat if you just go think about the last video when we're talking about normal normal vectors to a plane we can define a vector by we can define a plane by two vectors if we define a plane let's say that I have vector a right there and then I have vector B and let me do vector B like this those define a plane in r3 right let me define your plane so all the linear combinations of those two guys that's a plane in r3 all right you can kind of view it as they might form a subspace in r3 that forms a plane if you take a cross B you get a third vector that's orthogonal that's orthogonal to those two and so a cross B will pop out like this it'll be orthogonal to both of them and it look like that and so this vector right there is a cross B and you might say Sal how did you know that I mean there's multiple vectors that are orthogonal obviously the met the the length of the vector and I didn't specify that there but it could pop straight up like that or why didn't it you know you just as easily could have popped straight down like that that also would be orthogonal to a and B and the way that a cross B is defined you can essentially figure out the direction visually by using what's called the right hand rule and the way I think about it is you take your right hand take your right hand and let me see if I can draw a suitable right hand point your index finger in the direction of a so if your index finger is in the direction of a and then I point my middle finger in the direction of B so my middle finger in this case is going to go something like that my middle finger is going to do something like that and then my other fingers do nothing then my thumb my thumb will go in the direction of a cross B you could see that there my thumb is in the direction of a cross B and assuming that you are anatomically similar to me then you'll get the same result that let me draw it also this is vector a vector B goes in that direction then you know if you know hopefully you don't have a thumb hanging down here you know that you're that a cross B in this example will point up and it's orthogonal to both and just to make you you know to kind of satisfy you a little bit that the vector is definitely orthogonal or that this thing is definitely orthogonal to both of these let's just play with it and see that that definitely is the case and what is orthogonal what is the what is our really you know in our context the definition of orthogonal orthogonal vectors orthogonal if a if a and B are orthogonal that means that a dot B is equal to zero remember the difference you know orthogonal and perpendicular is that orthogonal also applies to zero vectors so these could also be zero vectors notice that I didn't say that any of these guys up here had to be nonzero well in a little bit we'll talk about the angle between vectors and then you have to assume nonzero but if you're just taking the cross product nothing to stop you from taking nothing no reason why any of these numbers can't be zero but let me show you let me show you that a cross B is definitely orthogonal to both a and B I think that might be somewhat satisfying to you so let me copy a cross B here I don't feel like rewriting it okay let me paste it okay but a little other stuff with it but let me take that with let me take the dot product of that with just my vector a which was just a 1 a 2 a 3 so what does the dot product look like it's that term times that so it's a 1 let me write get some space that's a 1 times a 2 B 3 a 2 B 3 minus a 1 times that minus a 1 times a 3 B 2 and then you have Plus this times this so plus a 2 times a 3 so plus a 2 times a 3 times B 1 and then minus minus a 2 a 2 A 1 B 3 and then finally plus I'll just continue it down here plus a 3 times a 1 B 2 minus a 3 times a 2 B 1 all I did is I just took the cross product of these two things right I just took each of this this guy times that was equal to those two terms that guy times that was equal to the next two terms equal to those two terms and then this guy times that was equal to those two terms and if these guys are really orthogonal then this should be equal to zero so let's see if that's the case so I have an A 1 a 2 B 3 a positive here and then I'm subtracting the same thing here right this is the same thing as a 1 a 2 B 3 but it's just a minus so that will cancel out with that and let's see what else we have we have a minus a 1 a 3 B - but we have a plus a 1 a 3 B - they're so these two are going to cancel out and I think you see where this is going you have a positive a 2 a 3 B 1 and then you have a negative a 2 a 3 B 1 there so these will also cancel out now I just showed you that it's orthogonal to a let me show you that it's orthogonal to be let me get another version of my of my the cross product of the two vectors if I scroll down a little bit and let me go back and let me multiply that times the vector B b1 b2 and b3 I'll do it here just have some space so b1 times this whole thing right here is b1 a2 b3 minus v1 times this minus b1 a3 b2 let me switch colors and then b2 times this thing here is going to be v2 and if so it's going to be plus this is all really one expression I'm just writing it on multiple lines this isn't a vector remember when you take the dot product of two things you get a scalar quantity so plus b2 times this thing so B 2 a 3 B 1 minus B 2 a 1 B 3 and then finally b3 times this so plus B 3 B 3 a 1 B 2 minus B 3 a 2 B 1 so if these guys are definitely orthogonal then this thing needs to equal zero and let's see if that's the case we have a B 1 a 2 B 3 so B 1 and a B 3 the B 1 a 2 B 3 that's a positive one and then this is a negative 1 you have a B 3 and a 2 and a B 1 so that and that cancel out here you have a minus B 1 a 3 B 2 so you have a b-1 and a B 2 you have a minus 1 it's minus B 1 a 3 B 2 this is a plus the same thing right that's just switching the order of the multiplication but these two are the same term they're just opposites of each other so they cancel out and then finally you have a B 2 and a 1 and a B 3 it's a negative and then you have a positive version of the same thing so these two guys cancel out so you see that this is also equal to 0 so hopefully you're satisfied that this vector right here is definitely orthogonal to both a and B and that's because that's how it was designed this is a definition you could do a little bit of algebra you could have you could have without me explaining this definition to you could have actually come up with this definition on your own but obviously this was kind of designed to have other interesting properties to it and I'll cover those in in the next few videos so hopefully you found that helpful