Vector dot and cross products
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Vector Dot Product and Vector Length
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Proving Vector Dot Product Properties
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Proof of the Cauchy-Schwarz Inequality
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Vector Triangle Inequality
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Defining the angle between vectors
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Defining a plane in R3 with a point and normal vector
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Cross Product Introduction
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Proof: Relationship between cross product and sin of angle
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Dot and Cross Product Comparison/Intuition
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Vector Triple Product Expansion (very optional)
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Normal vector from plane equation
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Point distance to plane
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Distance Between Planes
Proving Vector Dot Product Properties Proving the "associative", "distributive" and "commutative" properties for vector dot products.
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- 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.
Be specific, and indicate a time in the video:
At 5:31, how is the moon large enough to block the sun? Isn't the sun way larger?
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