Main content

### Course: Linear algebra > Unit 1

Lesson 5: Vector dot and cross products- Vector dot product and vector length
- Proving vector dot product properties
- Proof of the Cauchy-Schwarz inequality
- Vector triangle inequality
- Defining the angle between vectors
- Defining a plane in R3 with a point and normal vector
- Cross product introduction
- Proof: Relationship between cross product and sin of angle
- Dot and cross product comparison/intuition
- Vector triple product expansion (very optional)
- Normal vector from plane equation
- Point distance to plane
- Distance between planes

© 2024 Khan AcademyTerms of usePrivacy PolicyCookie Notice

# Vector dot product and vector length

Definitions of the vector dot product and vector length. Created by Sal Khan.

## Want to join the conversation?

- i still don't understand exactly, what IS a vector????(16 votes)
- More generally, a vector is just a collection of values. If the vector contains 3 values then we say that it's dimension is 3. In physics and geometry/trigonometry we talk about vectors having a magnitude and direction but you can also use vectors to hold other kinds of values. For example, if you were analyzing financial data, a vector might hold several characteristics of a company (e.g. Market Value, Number of Employees, Last Year Income, Last Year Profit, Number of years in business). See this video in the series:

http://www.khanacademy.org/video/linear-algebra--introduction-to-vectors

starting at time7:35to see the formal definition of a vector.(46 votes)

- How do you take the dot product of three vectors?(11 votes)
- You can't. When you take a dot product, it converts two vectors into a scalar. Attempting another dot product after that is impossible, because you would be trying to dot a scalar with a vector, which violates the definition of the dot product.(31 votes)

- is there any relation between the dot product of two vectors and cosine the angle between them?(9 votes)
- a · b = ‖a‖‖b‖cosθ so solving for the cosine of the angle θ:

cosθ = (a · b) / ‖a‖‖b‖**If a and b are unit vectors**then the denominator simplifies to 1, therefore cosθ = a · b.(21 votes)

- Why is the dot product a•b = a1b1+a2b2+...+anbn?

Is it just randomly difined as such or is there a proof that it has to be this?(11 votes)- I prefer to think of the dot product as a way to figure out the angle between two vectors.

If the two vectors form an angle A then you can add an angle B below the lowest vector, then use that angle as a help to write the vectors' x-and y-lengts in terms of sine and cosine of A and B, and the vectors' absolute values.

If you do that then you will end up with the equation**a·b**= |**a**|·|**b**|·cos(A).(9 votes)

- How do I find the component of a vector in a certain direction using dot product?(6 votes)
- This is where projections come in. If you have a vector a that you want to project onto a direction given by vector b, you use (a · b)b/(b · b) [that is a dot b over b dot b all multiplied by b]. Note that b · b is the square length of b, so we can write (a · b)b/b², or some might prefer the notation (a · b)b/||b||².

Later on in the Linear Algebra playlist, Sal goes through projections. I can't recommend this MIT lecture highly enough for projections though: http://www.youtube.com/watch?v=Y_Ac6KiQ1t0(11 votes)

- How is it possible for two quantities with direction to have a product that does not have one? If we take two vectors and multiply them using both dot and cross methods then will the solutions vary? Or should we use dot and cross products for different situations? If so, how do we figure out?(2 votes)
- You shouldn't think of the dot and cross product as "multiplication". Taking a dot product is taking a vector, projecting it onto another vector and taking the length of the resulting vector as a result of the operation. Simply by this definition it's clear that we are taking in two vectors and performing an operation on them that results in a scalar quantity.

We could have defined an operation that takes in two vectors and returns`true`

if they are perpendicular,`false`

otherwise. Then you'd have an operation that takes in two vectors and results in a boolean value. My point being that you can take any parameter types and return any desired type independently. I don't see how the types are related!(10 votes)

- What can you use the dot product for?(5 votes)
- One application is detecting orthogonality/perpendicularity of vectors if the dot product is zero.

If you divide the dot product by the lengths of the vectors, you're left with the cosine of the angle between the vectors, so the arc cosine of this will be the angle.(4 votes)

- This in no way explains what the dot product is. It gives you a number, sure, great, but what does that number mean? I looked it up elsewhere and am leaving this comment: The dot product of two vectors a and b, is equal to the product of the following three terms, (the length of a) squared, (the length of b) squared, and cosine theta. Where theta represents the angle between a and b. If you have 2 vectors, you could use the dot product to determine the angle between them.(6 votes)
- That definition is in this video, and he actually derived it on one of the other videos using the law of cosines.(1 vote)

- Can you reduce the problem to the A-vector in the brakets squared, equal to simple A-vector ^2(5 votes)
- Wait. This is a college level course? At4:56?(3 votes)
- Linear algebra is usually taken at the early college level, yes. There is some intersection with precalculus, like in the more basic properties of vectors, but the bulk of this is college material.

That said, you really don't need much in the way of prerequisites to study linear algebra. Even though it's often taken after calculus, you don't need to know calculus to study linear algebra.(2 votes)

## Video transcript

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 b. 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 multiplication. 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. OK. 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 components. 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 definition. 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.