Vector dot product and vector length Definitions of the vector dot product and vector length
Vector dot product and vector length
- We've already made a few definitions of operations that
- we can do with vectors.
- We've defined addition in the context of vectors
- and you've seen that.
- If you just have two vectors, say a1, a2, all the
- way down to a n.
- We defined the addition of this vector and let's say some
- other vector, b1, b2, all the way down to
- bn as a third vector.
- If you add these two, we defined the addition operation
- to be a third-- you will result in a third vector where
- each of its components are just the sum of the
- corresponding components of the two vectors you're adding.
- So it's going to be a1 plus b1, a2 plus b2, all the way
- down to a n plus bn.
- We knew this and we've done multiple videos where we use
- this definition of vector addition.
- We also know about scalar multiplication.
- Maybe we should just call it scaling multiplication.
- And that's the case of look, if I have some real number c
- and I multiply it times some vector, a1, a2, all the way
- down to a n, we defined scalar multiplication of a vector to
- be-- some scalar times its vector will result in
- essentially, this vector were each of its components are
- multiplied by the scalar.
- ca1, ca2, all the way down to c a n.
- And so after seeing these two operations, you might be
- tempted to say, gee, wouldn't it be nice if there was some
- way to multiply two vectors.
- This is just a scalar times a vector, just scaling it up.
- And that's actually the actual effect of what it's doing if
- you visualize it in three dimensions or less.
- It's actually scaling the size of the vector.
- And we haven't defined size, very precisely just yet.
- But you understand at least this operation.
- For multiplying vectors or taking the product, there's
- actually two ways.
- And I'm going to define one of them in this video.
- And that's the dot product.
- And you signify the dot product by saying a dot v.
- So they borrowed one of the types of multiplication
- notations that you saw, but you can't write across here.
- That'll be actually a different type of vector
- So the dot product is-- it's almost fun to take because
- it's mathematically pretty straightforward, unlike the
- cross product.
- But it's fun to take and it's interesting because it
- results-- so this is a1, a2, all the way down to a n.
- That vector dot my b vector: b1, b2, all the way down to bn
- is going to be equal to the product of each of their
- corresponding components.
- So a1 b1 added together plus a2 b2 plus a3 b3 plus all the
- way to a n, bn.
- So what is this?
- Is this a vector?
- Well no, this is just a number.
- This is just going to be a real number.
- You're just taking the product and adding together a bunch of
- real numbers.
- So this is just going to be a scalar, a real scalar.
- So this is just going to be a scalar right there.
- So in the dot product you multiply two vectors and you
- end up with a scalar value.
- Let me show you a couple of examples just in case this was
- a little bit too abstract.
- So let's say that we take the dot product of the vector 2, 5
- and we're going to dot that with the vector 7, 1.
- Well, this is just going to be equal to 2 times 7 plus 5
- times 1 or 14 plus 6.
- No, sorry.
- 14 plus 5, which is equal to 19.
- So the dot product of this vector and this vector is 19.
- Let me do one more example, although I think this is a
- pretty straightforward idea.
- Let me do it in mauve.
- Say I had the vector 1, 2, 3 and I'm going to dot that with
- the vector minus 2, 0, 5.
- So it's 1 times minus 2 plus 2 times 0 plus 3 times 5.
- So it's minus 2 plus 0 plus 15.
- Minus 2 plus 15 is equal to 13.
- That's the dot product by this definition.
- Now, I'm going to make another definition.
- I'm going to define the length of a vector.
- And you might say, Sal, I know what the length
- of something is.
- I've been measuring things since I was a kid.
- Why do I have to wait until a college level or hopefully
- you're taking this before college maybe.
- But what is now considered a college level course to have
- length defined for me.
- And the answer is because we're abstracting things to
- well beyond just R3 or just three-dimensional space is
- what you're used to.
- We're abstracting that these vectors could have 50
- And our definition of length will satisfy, will work, even
- for these 50 component vectors.
- And so my definition of length-- but it's also going
- to be consistent with what we know length to be.
- So if I take the length of a and the notation we use is
- just these double lines around the vector.
- The length of my vector a is equal to-- and this is a
- It equals the square root of each of the terms, each of my
- components, squared.
- Add it up.
- Plus a2 squared plus all the way to plus a n squared.
- And this is pretty straightforward.
- If I wanted to take let's call this was vector b.
- If I want to take the magnitude of vector b right
- here, this would be what?
- This would be the square root of 2 squared plus 5 squared,
- which is equal to the square root of-- what is this?
- This is 4 plus 25.
- The square root of 29.
- So that's the length of this vector.
- And you might say look, I already knew that.
- That's from the Pythagorean theroem.
- If I were to draw my vector b-- let me draw it.
- Those are my axes.
- My vector b if I draw it in standard
- form, looks like this.
- I go to the right 2.
- 1, 2.
- And I go up 5.
- 1, 2, 3, 4, 5.
- So it looks like this.
- My vector b looks like that.
- And from the Pythagorean theorem you know look, if I
- wanted to figure out the length of this vector in R2,
- or if I'm drawing it in kind of two-dimensional space, I
- take this side which is length 2, I take that side which is
- length 5; this is going to be the square root from the
- Pythagorean theorem of 2 squared plus 5 squared.
- Which is exactly what we did here.
- So this definition of length is completely consistent with
- your idea of measuring things in one-, two- or
- three-dimensional space.
- But what's neat about it is that now we can start thinking
- about the length of a vector that maybe has 50 components.
- Which you know, really to visualize it in our
- traditional way, makes no sense.
- But we can still apply this notion of length and start to
- maybe abstract beyond what we traditionally
- associate length with.
- Now, can we somehow relate length with the dot product?
- Well what happens if I dot a with itself?
- What is a dot a?
- So that's equal to-- I'll just write it out again.
- a1, all the way down to a n dotted with a1 all the way
- down to a n.
- Well that's equal to a1 times a1, which is a1 squared.
- Plus a2 times a2.
- a2 squared.
- Plus all the way down, keep doing that all the way down to
- a n times a n, which is a n squared.
- But what's this?
- This is the same thing as the thing you
- see under the radical.
- These two things are equivalent.
- So we could write our definition of length, of
- vector length, we can write it in terms of the dot product,
- of our dot product definition.
- It equals the square root of the vector dotted with itself.
- Or, if we square both sides, we could say that our new
- length definition squared is equal to the dot product of a
- vector with itself.
- And this is a pretty neat-- it's almost trivial to
- actually prove it, but this is a pretty neat outcome and
- we're going to use this in future videos.
- So this is an introduction to what the dot product
- is, what length is.
- In the next video I'm going to show a bunch of
- properties of it.
- It will almost be mundane, but I want to get all those
- properties out of the way, so we can use
- them in future proofs.
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At 5:31, how is the moon large enough to block the sun? Isn't the sun way larger?
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