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Current time:0:00Total duration:18:09

Proof: Relationship between cross product and sin of angle

Video transcript

the goal of this video is to start with our definition of the cross product and the result that we started off with in or that we got to in a different video I think it was three videos ago where we found out that the dot product of two nonzero vectors a dot B is equal to the product of their lengths so the product of the length of a width the length of B times the cosine of the angle between them we're going to start with these two things this definition of a cross product in r3 the only place it really is defined and then this result and we want to get to we want to get to the result that the length the length of the cross product of two vectors and so obviously when you take a cross product you get a vector but if you take its length you get a number again you just get a scalar value is equal to the product of each of the vectors lengths so the product of the length of a time's the product of the length of B times the sine of the angle between them which is a pretty neat outcome because it kind of shows that there are two sides of the same coin dot product has cosine cross product has signed I'm sure you've seen this before and in in in well you you definitely have seen it if you watch my physics playlist and I even do a whole video or I talk about the intuition by behind what this really means and I encourage you to rewatch that and I'll probably do that again in the linear algebra context but the point of this video is to prove this to you is to prove that with this and this I can get to this now if you just believe me and you just say oh I've seen that before and I just think it's definitely is the case then you don't have to watch the rest of this video because I'll tell you right now it's gonna get dirty it's gonna be a hairy hairy proof but if you're willing to watch and bear with me well let's let's start start on this start on this start proving this result so the place I'm gonna start is with the idea of taking the length of a cross B squared that's a cross B right there so if I'm essentially taking the length of this vector squared and we saw in many videos and I've used this idea multiple times but if I just have some arbitrary vector let me just say some arbitrary vector and I take its length squared that's just equal to that vector dotted with itself or the square of each of its terms summed up all the way to xn squared so what will this be equal to well this is just equal that vector and we only have three components so it's equal to the sum of the squares of each of these components so it's equal to let me write this down it's equal to this term squared so let me write that down a2 b3 minus a3 b2 squared plus this term squared so plus a3 b1 minus a1 b3 squared and then finally plus that term squared so plus a1 b2 minus a2 b1 squared and what does this equal to well let's just expand it out let's expand that out so this this term right here we're just gonna have to do our our expansion of the square of a binomial and we've done this multiple times so this is going to be equal to a2 squared b3 squared and then we're gonna have these two multiplied by each other twice so minus two I'm just I'm just multiplying this out minus 2 times a2 a3 b2 b3 I'm just rearranging them to get the order right plus a3 squared b2 squared that term squared and then I'll have then I have to add this term so plus a3 squared a3 squared b1 squared minus 2 times both of these terms multiplied minus 2 times a1 a3 b1 b3 plus that term squared a 1 squared b3 squared and then finally this term squared so plus plus a 1 squared B 2 squared minus 2 times a 1 a 2 B 1 B 2 plus a 2 squared B 1 squared so there you go and let's see if we can write this in a form well I'm going to write this in a form that I know will be useful later so what I'm gonna do is I'm gonna factor out the a a 2 A 1 a 3 squared terms so I could write this as let me pick a new neutral color so this is equal to if I just write a 1 squared where's my a 1 squared terms I got that one right there and I have that one right there so a 1 squared times B 2 squared plus B 3 squared plus B 3 squared good enough now we're my a 2 squared terms a plus a 2 squared times I have that one and that one so times B 1 squared that's that plus B 3 squared plus B 3 squared and then finally let me pick another new color well go back to yellow plus a 3 squared times well that's that term in that term so B 1 and B 2 so B 1 squared plus B 2 squared and obviously I can't forget about all of that mess that I have in the middle all of this stuff right here all of that stuff right there so plus or maybe I should write minus 2 minus 2 times all of this stuff let me just write it real fast so it's a 2 a 3 B 2 B 3 plus a1 a3 b1 b3 plus a 1 a 2 A 1 a 2 B 1 B 2 there you go now let's put this aside for a little bit let me put this let me put this on the side for a little bit we'll let that equation rest for a little while and remember this is just an expansion of the length of be squared that's all this is so just remember that and now let's do another equally hairy and cumbersome computation let's take this result up here we know that we know that the magnitude or the length of a time's the length of B times the angle between them is equal to a dot B which is the same thing as if we actually do the dot product a 1 times B 1 plus a 2 times B 2 plus a 3 times B 3 now four just to kind of make sure that you know I get to do the hairiest problem possible let's take the square of both sides so let's take a square this side you get a squared B squared cosine squared then you got a dot B squared or you get hold this thing hold this whole thing squared so what's this whole thing squared for me it's easier to just write out the thing again instead of writing a square just multiply that times a 1 B 1 plus a 2 B 2 plus a 3 B 3 and let's do some let's do some polynomial multiplication so first let's multiply this guy times each of those guys so you have a 1 B 1 times well there a 1 B 1 so you get I'm gonna do it right here you get a 1 squared B 1 squared plus a 1 plus this guy times this guy plus a 1 a 2 a 1 a 2 times B 1 B 2 Plus this guy times that guy plus a 1 a 3 times B 1 B 3 fair enough now let's do the second term we have to multiply this guy times each of those guys so a 2 B 2 times a 1 B 1 well that's this one right here a 2 a 2 B 2 times a 1 B 1 a be one I wrote it right here because this is really the same term and eventually we want to simplify that so that's that times that guy then we have this guy times that over there so let me write it over here so that's a 2 squared be 2 squared put a plus right there and then finally this middle guy times this third guy so let me write it over here plus plus so a 2 a 2 a 3 B 2 B 3 now we only have one left we only have one left and I'll do it in maybe I'll do it in this blue color I have to multiply this guy times each of those guys so a 3 B 3 times a 1 B 1 that's the same thing as this term right here right because you have a 3 yeah if let me write it right here you have a 3 B 3 times a 1 B 1 a 1 B 1 then you have this guy times that guy which is this because it's a 3 a 3 B 3 times a 2 a 2 B 2 let me put a little plus sign there and then finally you have this guy times himself so you have a 3 squared B 3 squared and so if you add up all of this business here what do you get what do you get you get I'll switch to another color you have a 1 squared B 1 squared plus and I'm doing these colors in a certain way on purpose plus a 2 squared B 2 squared plus plus a 3 squared B 3 squared plus and let me do it in this I'll do it in white plus what do you have here you have this term times 2 you have this term times 2 and then you have this term times 2 so plus 2 times a 1 let me write that down plus 2 times a1 a2 b1 b2 that's that term plus this one right here plus a1 a3 b1 b3 finally plus this 1 a2 a3 plus b2 b3 and you might have noticed something interesting already if you compare this term right here if you compare that guy right there to this guy right there they're the same thing you have an a1 a2 b1 b2 a1 a2 b1 b2 this term and that term are the same let's look at the other terms let me pick a nice color a 1 a 3 B 1 B 3 a 1 a 3 B 1 B 3 that term and that term is the same and then finally if you compare a 2 a 3 B 2 B 3 or this shouldn't be a plus this is just this one so a 2 a 3 this one that's just they're all multiplied a 2 a 3 B 2 B 3 a 2 a 3 B 2 B 3 this term and this term is the same and this expression when we expanded it out we have 2 times this positive 2 times this and this term right here when we expanded it out we have minus 2 times this so you might see so let's see if we can simplify things a little bit so what happens what happens if we add this guy to this guy let's do it let's do it so it's a little exciting so we get a cross B the length of that squared we're going to add to that this expression right here so plus the length of a squared times the length of B squared times the cosine of the angle between them squared what's that going to be equal it's going to be equal to this thing Plus this thing and let's do a simplification what's this thing Plus this thing well we already said that this is the minus 2 times this this is the plus 2 times this so this guy let me be very clear this right here is going to cancel out when we add the two terms is going to cancel out with this guy these guys are going to cancel out thank God cancel cancel out makes our life a little bit easier and what are we left with we're left with we're left with this right here Plus that right there then we see we have an a.1 squared term so we just add the coefficients on the a one squared we add the coefficients of the a two squared and we add the coefficients on the a three squared and what do we get we get we get a 1 squared a 1 squared times this coefficient plus this coefficient so you get B 1 squared plus B 2 squared plus B 3 squared things are starting to look a little bit orderly all of a sudden and then you have plus a 2 squared times the all of their coefficients add it up so B 1 squared plus B 2 squared plus B 3 squared and then finally in yellow you have plus a 3 sorry I'm going to trying to do that in yellow you have a 3 squared and you have that you have B 1 squared B 2 squared and B 3 squared so B 1 squared plus B 2 squared plus B 3 squared and if you see we're multiplying all of these things by this B 1 squared plus B 2 squared plus B 3 squared so we can actually factor that out and we get something very interesting so this is equal to if we factor the B what that this thing out of all the terms we get B 1 squared plus B 2 squared plus B 3 squared times my a squared terms times a 1 squared plus a 2 squared plus a I'm getting excited the homestretch is here a 3 squared so these two things are equal to each other but what's this thing what's another way I could write this this is the same thing as B dot B or the length of my vector B squared and what's that that's the length of my vector a squared this is my length of my vector a squared that's just a dot a so we have let me write rewrite everything so we have the length of a it's a darker green egg cross B squared plus this thing plus the length of and I wanted a Plus this let me actually just copy and paste it it's monotonous plus that thing right there now why isn't it if I control copy and paste so it's not working all right so Plus that thing a the length of a squared times the length of B squared times the cosine of the angle squared between them is equal to that now what if we subtract what if we subtract this from both sides all right what do we get we get the length of a cross B squared is equal to this minus this and we can factor so let me write that so actually let me just subtract this on this line so if I subtract it from both sides I could get that out there and I'll put the minus the length of a squared times the length of B squared times the cosine squared of the angle between them and we can factor this a squared B squared that's the lengths of they're the two vectors out right I'm just switching the order so this is equal to the length of a squared times the length of B squared times and this is this is exciting times this when you factor this out of this you just get a 1 minus cosine squared of theta and what is 1 minus cosine squared of theta well sine squared of theta plus cosine this is the most basic trig identity sine squared of theta plus cosine squared of theta is equal to 1 so if you subtract cosine squared from both sides you get sine squared of theta is equal to 1 minus cosine squared of theta so this is this is sine squared of theta and then what happens if you take the square root of both sides and this is really exciting you get the length of vector a crossed with vector B is equal to the length of vector a times the length of vector B times the sine of the angle between them right I just took the square root of both sides of this and we finally get our result I never thought I would get here and so hopefully you're satisfied you never have to take this as kind of a leap of faith anymore and hopefully you're satisfied with this and I'm going to stop recording this video before I make a careless mistake or the power goes out that would ruin everything