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Current time:0:00Total duration:10:45

Proving vector dot product properties

Video transcript

in this video I want to prove some of the basic properties of the dot product and you might find what I'm doing in this video somewhat mundane but you know to be frank it is somewhat mundane but I'm doing it for two reasons one is this is the type of thing that's often asked of you and when you take a linear algebra class but more importantly it gives you the appreciation that we really are kind of building up a mathematics of vectors from the ground up and you really can't assume anything you really need to prove everything for yourself so the first thing I want to prove is that the dot product that when you take the vector dot products if I take V dot W that is commutative that the order that I take the dot product doesn't matter I want to prove to myself that that is equal to W dot V and so how do we do that well and this is the general pattern for a lot of these vector proofs let's just write out the vectors so V will look like v1 v2 all the way down to VN let's say that this is equal to V and let's say that W is equal to w1 w2 all the way down to W n all the way down to W a so what is V dot W equal V dot W is equal to I'll switch colors here v1 times w1 whoops v1 w1 plus v2 w2 plus all the way to V and W n fair enough now what does what does W dot V equal well WV you know when I've made the definition it's you know you just multiply the products but I'll just do it in the order that they gave it to us so it equals W 1 v1 plus W 2 v2 plus all the way to W and VN now these are clearly equal to each other but because if you just match up the first term with the first term those are clearly equal to each other V 1 W 1 is U - w1 v1 and I can say this now because now we're just dealing with regular numbers here we were taking we were doing with vectors and we were taking this weird type of multiplication called the dot product but now I can definitely say that these are equal because this is just regular multiplication and this is just a commutative property commutative let me see if I'm spelling commutative commutative property of multiplication we learned this and you know I don't know when you learn this and second or third grade so you know that those are equal and by the same argument you know that these two are equal then you could just rewrite each of these terms just by switching that around and that's just from the community from basic multiplication of scalar numbers of just regular real numbers so that's what tells us that these two things are equal or these two things are equal so we've proven to ourselves that order doesn't matter when you take the dot product now the next thing we could take a look at is whether the dot product exhibits the distributive property so let me just define another vector X here another vector X and you can imagine how I'm going to define it x1 x2 all the way down to X n now what I want to see if dot product the dot product deals with the distributive property the way I would expect it to then if I were to add V plus W and then multiply that by X by the by and first of all it shouldn't matter what order I do that with I just showed it here I could do X dot this thing it shouldn't matter because I just show you is commutative but if the distribution works then this should be the same thing as V dot X plus W dot X right if these were just numbers this is just regular multiplication you would multiply it by each of the terms and that's what I'm showing here so let's see if this is true if this is true for the dot product so what is V plus W V plus W V plus W is equal to is equal to we just add up each of their corresponding terms V 1 plus W 1 V 2 plus W 2 all the way down to VN plus WN that's that right there and then when we dot that when when you dot that with X 1 X 2 all the way down to X n what do we get well we get V 1 plus W 1 times X 1 plus V 2 plus W 2 times X 2 plus all the way to VN plus W n times X n all right I just took the dot product of these two I just multiplied corresponding components and then added them all up that was the dot product this is V plus W dot X let me write that down this is V plus W dot X now let's work on these things up here what is V dot let me write it over here what is V dot X V dot X we've seen this before this is just V 1 X 1 oh no no vectors now these are just their actual components plus V 2 X 2 all the way to VN X n what is WX WX is equal to W 1 X 1 plus W 2 X 2 all the way to W n X n now what is what do you get when you add these two things and not notice I'm just adding here I'm adding two scalar quantities that's a scalar that's a scalar we're not doing vector addition anymore so this is a scalar Connie and this is a scalar quantity so what I get when I add them so V dot X plus W X is equal to V 1 X 1 plus W 1 X 1 plus V 2 X 2 plus W 2 X 2 all the way to VN x and plus W and X and I know it's very monotonous but you could immediately see we're just dealing with regular numbers here so we can take the X's out and what do you get this is equal to 1 right here this is equal to we could just take the X out factor the X out V 1 plus W 1 X 1 plus V 2 plus W 2 X 2 all the way to VN plus WN X n which we see this is the same thing as as this thing right here as this thing right here so we just showed that this expression right here is the same thing as that expression or the distribution the distributive property seems to or does apply and the way we would expect to the dot product and I know you're so mundane why are we doing but I'm doing this because show you that we're building things that we couldn't just assume this but the proof is pretty straightforward and in general I didn't do these proofs when I did it for vector addition and scalar multiplication and I really should have but you can prove the commutativity of it or for the scalar multiplication you could prove that distribution work for it doing a proof exactly the same way is this a lot of math books or linear algebra books just leave these as exercises to the student took because it's mundane so they didn't think it was worth their paper but let me just show you I guess the last property let me just say associativity the associative property that doesn't matter where so let me let me show you if I take some scalar and I multiply it times V some vector V and then I take the dot product of that with W if if this is associative the way multiplication in our everyday world normally works this should be equal to and it's still a question mark because I haven't proven it to you it should be equal to C times V dot W so let's figure it out what's C times the vector V C times the vector V is C times V 1 C times V 2 all the way down to C times V N and then the vector W we already know what that is so dot W is equal to what it's equal to this times the first term of W so C v1 w1 Plus this times the second term of w sze v2 w2 all the way to C n WN fair enough that's what this side is equal to now let's do this side what is V dot W I'll write it here V dot W done this multiple times this is just V 1 W 1 plus V 2 W 2 all the way to VN WN I'm getting tired of doing this and you're probably tired of watching it but it's good to go through the exercises you know someone asked you to do this now you'll be able to do this now what is C times this so if I multiply some scalar times this that's the same thing as multiplying some scalar times that so now I'm just multiplying a scalar times a big this is just the regular distribute distributive property of just numbers of just regular real numbers so this is going to be equal to C V 1 W 1 plus C V 2 W 2 plus all the way to C V n WN and we see that this is equal to this because this is equal to this now the hardest part of this I remember when I first took linear algebra I found when when the professor would assign assign you know prove this I would have trouble doing it because it almost seems so ridiculously obvious that hey well you know obviously if you just look at the components of them they just it just turns into multiplying of each individual components and adding them up and those are associative so that's obviously so what what's there to prove and it only took me a little while that they just wanted me to write that down this you know they didn't want something earth-shattering they just wanted me to show when you take go component by component and all you have to do is associate is assume you know the kind of distributive or the associative or the commutative property of regular numbers that you could prove these the same properties also apply in a very similar way to vectors and the dot product so hopefully you found this reasonably useful and I'll see you in the next video where we could use some of these tools to actually prove some more interesting properties of vectors