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. 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 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 ready to prove everything

for yourself. So the first thing I want to

prove is that the dot product, when you take the vector dot

product, so if I take v dot w that it’s 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 wn. So what does v dot w equal? v dot to w is equal to–

I’ll switch colors here– v1 times w1. Plus v2 w2 plus all

the way to vn wn. Fair enough. Now what does w dot v equal? Well w dot v– you know, when I

had made the definition, you just multiply the products. But I’ll just do it in the order

that they gave it to us. So it equals w1 v1 plus w2 v2. Plus all the way to wn vn. Now, these are clearly equal to

each other because if you just match up the first term

with the first term, those are clearly equal to each other.

v1 w1 is equal to w1 v1. And I can say this now because

now we’re just dealing with regular numbers. Here we were dealing 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. Let me see if I’m spelling

commutative. We learned this in– I don’t

know when you learned this, in second or third grade. So you know that those are equal

and by the same argument you know that these

two are equal. You could just rewrite each of

these terms just by switching that around. That’s just 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 xn. Now, what I want to see if 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. 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 showed you it’s commutative. But if the distribution works,

then this should be the same thing as v dot x plus w dot x. If these were just numbers

and this was just regular multiplication, you would

multiply by it by each of the terms, and that’s what

I’m showing here. So let’s see if this is true

for the dot product. So what is v plus w? v plus w is equal to– we just

add up each of their corresponding terms. v1 plus

w1, v2 plus w2, all the way down to vn plus wn. That’s that right there. And then when we dot that with

x1, x2, all the way down to xn, what do we get? Well we get v1 plus w1 times

x1 plus v2 plus w2 times x2 plus all the way to vn

plus wn times xn. 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. Let me write it over here. What is v dot x? v dot x, we’ve seen

this before. This is just v1 x1. No vectors now. These are just actual

components. Plus v2 x2, all the

way to vn xn. What is w dot x? w dot x is equal to w1 x1 plus

w2 x2, all the way to wn xn. Now what do you get when you

add these two things? And notice, 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 quantity and

this is a scalar quantity. So what do I get when

I add them? So v dot x plus w dot x is equal

to v1 x1 plus w1 x1 plus v2 x2 plus w2 x2, all the

way to vn xn plus wn xn. 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? Let me write it here. This is equal to– we could

just take the x out, factor the x out. v1 plus w1, x1 plus v2 plus

w2 x2, all the way to vn plus wn xn. Which we see this is

the same thing 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 the way we would expect to the dot product. I know this is so mundane. Why are we doing this? But I’m doing this to show you

that we’re building things up. 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 works for it

doing a proof exactly the same way as this. A lot of math books or linear

algebra books just leave these as exercises to the student

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, associativity, the associative

property. So 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 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

v1, c times v2, all the way down to c times vn. 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, c v2 w2, all the way to c vn 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. We’ve done this multiple

times. This is just v1 w1 plus v2

w2, 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, if 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 I’m just multiplying a scalar

times a big– this is just the regular distributive

property of just numbers, of just regular real numbers. So this is going to be equal to

c v1 w1 plus c v2 w2 plus all the way to c vn 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

the professor would assign, you know, prove this. I would have trouble doing it

because it almost seems so ridiculously obvious. That hey, well, obviously

if you just look at the components of them, it just

turns into multiplying of each individual component and adding

them up and those are associative, so that’s

obviously– what’s there to prove? And it only took me a little

while that they just wanted me to write that down. They didn’t want something

earth shattering. They just wanted me to show

when you go component by component and all you have to

do is assume kind of the distributive or the associative

or the commutative property of regular numbers,

that you could prove 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.

Great video, keeeeeeeeeeeeeeep coming i love it.

Good video. I find this sort of stuff difficult because it's so basic and low level. When going through a maths book from start to finish, the chapter on proof was one of the most difficult for me.

fell asleep >.<

can you prove VdotW= |V||W| cos @????

You really helped me man!

Thanks from Brasil! 🙂

Thank you! I HATE linear algebra. You don't know how much frustration that saved me.

DOT PRODUCT

A.B = AB cos (angle) means that the net force is a combination of

1] Force of A

2] Force of B

3] Angle at which A and B are clashing into each other.

Cos is a measure of how parallel two forces are, hence cos(0) = maximum =1.

And Cos comes in mathematics wherever two objects produce maximum impact when they are parallel to each other.

There is a video on this on my channel. Hope this helps.

BINNOY

Visualizing Maths(moving beyond formulas)

Available at Amazon and Kindle for 4$

anybody know the program he is using? thanks

why the column matrix can be multiplied in this way? at time 2.01, first column matrix is 3*1 and second column matrix is 3*1 ? can matrix multiplied like that?

Physics exam tomorrow! Thanks!

your explanation is so wonderful but the video would be completely usefull if you would have added some examples

Thanks but this was not helpful 🙁 My professor wants us to write down to the side properties which allows us to take each individual steps such as Multiplication is Commutative , Distributive property, etc. For example if we move the parenthesis we need to write on the side why we can do that so we would write ASSOCIATIVITY.

you are right it is too mundane.

7:38 the truth

how come he doesn't use sigma notation instead of writing

…all the time?dot product is defined as A B cos(angle)

and you just said in the beginning of the video we will build every thing up from simple maths

how could you define A (dot) B like that ?

What about <V,V> >= 0 & <V,V> = 0 if and only if v=0 AKA Positivity Axiom????

Very fantastic indeed . Thank you

Hey Khan academy,

In this video,

aren't we using the distributive law to prove the distributive law

Thank you kind sir, for teaching us how to understand math, not how to remember math formulas. 10/10 teacher, would study again !

9:56 literally me