Main content

## Physics library

# Dot vs. cross product

This passage discusses the differences between the dot product and the cross product. While both involve multiplying the magnitudes of two vectors, the dot product results in a scalar quantity, which indicates magnitude but not direction, while the cross product results in a vector, which indicates magnitude and direction. The author also explains how to use the right hand rule to determine the direction of the cross product vector. Created by Sal Khan.

## Want to join the conversation?

- Why are they multiplied though? What does |a||b|cos(theta) really represent conceptually? I see what they are and what the difference between them is, but I don't see why they should be multiplied rather than, say, added.(30 votes)
- Hi John, when dealing with vectors, the orientation in the coordinate system and among each other are decisive factors, that you don't have when multiplying plain scalar numbers.

With the two kinds of multiplication of vectos, the projection of one to the other is included.

Taking, for example, two parallel vectors: the dot product will result in cos(0)=1 and the multiplication of the vector lengths, whereas the cross product will produce sin(0)=0 and zooms down all majesty of the vectors to zero.

Another difference is the result of the calculation: Sal showed, that you're getting a plain SCALAR (number) as a result of the dot product, whereas the cross product only really makes sense in the 3d-space, because the resulting VECTOR is perpendicular to both other vectors (right hand rule). Try that in 2 dimensions ;-)(28 votes)

- can anyone please tell me the difference between precision and accuracy?(2 votes)
- Accuracy is about how close you get to the true value; if the actual answer is 4 and you get 3.9 then that's more accurate than 9.6; it's closer to 4. Precision is to do with the spread of your values; the more closely grouped the values you get, the more precise they are. For example, values of 8.9, 8.8, 8.9, 8.7, 8.8 are more PRECISE than 3.6, 4.7, 5.3, 2.6, 4.2 but the second set of values are more ACCURATE as they are closer to 4 on average.

I hope this made some sense to you!(23 votes)

- That was very useful, thank you. But if i may ask, when do i use the dot product and when do i use the cross product? they almost have opposite meanings. another thing In physics when we multiply 2 forces we just, for example do 10X8 and that's it. and what exactly is the vector projection, is it the "shadow" as you referred to it?(6 votes)
- why if 2 vectors perpendicular to each other are crossed do I get a vector orthogonal to both of em?? maybe because, if I were compressing two strong steel rods mutually perpendicular with a supermassive force, they would rather bend into the 3rd dimension tha the two forces' resultant :D(5 votes)
- A good example is gyroscopic force. Vectors that work together to cause a disk to spin create an orthoganal force that resists change.(1 vote)

- Do you, after explaining the cross product, you just use a determinant as a shortcut in later videos? That would be very simple to use for later, once you know the actual reasoning.(4 votes)
- I shall be optimistic and ask: To what shortcut do you refer?

The determinant I know requires the same number of steps because it computes the same function. So let's define the cross product as a determinant, right now.

{Wave magic wand}

Am I missing a shortcut? What step can I leave out?

I'm not making fun. Such a shortcut would be very useful to me. I have hunted for one. But I think the answer is in the question; the determinant constructs each term by definition. No mathematical shortcuts. Am I missing something?(1 vote)

- While finding the dot product, why is it that the final quantity doesn't have any direction?(2 votes)
- I learned in school about a different method of the dot product. That is:

Two vectors- x <a1,b1> and y <a2, b2>

and the dot product of x dot y was a1a2+b1b2

Is this also another correct way to do the dot product?

Thanks(2 votes)- Oh okay, Thank you for clarifying!(1 vote)

- Is the product of two similar vectors(say, velocity) always a dot product?

I mean, can a 'vector a' when multiplied with itself yeild a cross product?(2 votes)- It won't. The result would be zero, since the sine of 90 degrees is zero.(1 vote)

- If a is opposite in direction to b, in which direction will the n be in? I tried to do it with the right hand rule but it can go both ways.(1 vote)
- It´s actually 0, the direction is the same, you just change the orientation.

The more rigorous way to see it is with the sine , sin(0)=0 and sin(180º) are both 0, so operating you get that n is 0, no direction.(2 votes)

- Why doesn't the dot product of two vectors give us a vector?(1 vote)
- because that's the definition of the dot product. It's also called the scalar product.(2 votes)

## Video transcript

Let's do a little compare and
contrast between the dot product and the cross product. Let me just make two vectors--
just visually draw them. And maybe if we have time,
we'll, actually figure out some dot and cross products
with real vectors. Let's call the first one--
That's the angle between them. OK. So let's just go over the
definitions and then we'll work on the intuition. And hopefully, you have a little
bit of both already. So what is a dot b? Well first of all, that's the
exact same thing as b dot a. Order does not matter when you
take the dot product because you end up with just a number. And that is equal to the
magnitude of a times the magnitude of b times cosine
of the angle between them. Let's look at the definition
of the cross product. What is a cross b? Well first of all, that does
not equal b cross a. It actually equals the opposite
direction, or you could view it as the negative
of b cross a. Because the vector that you end
up with ends up flipped, whichever order you do it in. But a cross b, that is equal to
the magnitude of vector a times the magnitude of vector
b-- so far, it looks a lot like the dot product, but this
is where the diverge is-- times the sine of the
angle between them. The sine of the angle
between them. And this is where it
really diverges. When we took the dot
product, we just ended up with a number. This is just a number. There's no direction here. This is just a scalar
quantity. But the cross product, we take
the magnitude of a times the magnitude of b, times the sine
of the angle between them, and that provides a magnitude, but
it also has a direction. And that direction is provided
by this normal vector. It's a unit vector. A unit vector gets that
little hat on it. It's a unit vector, and
what direction is it? Well, that's defined by
the right hand rule. This is a vector. It's perpendicular
to both a and b. And then you might say, a and
b, the way I drew them, they're both sitting in the
plane of this video screen, or your video screen. So in order for something to
be perpendicular to both of these, it either has to pop out
of the screen or pop into the screen, right? And when you learned about the
cross product, I said there's two ways of showing a vector
popping out of the screen. It looks like that because
that's the tip of an arrow. And to show a vector going into
the screen, it's like that because that is the
back of an arrow. The rear end of an arrow. So how do you know which
of these two it is? Because both of these vectors
are perpendicular to a and b. That's where you take your right
hand and you use the right hand rule. So you take your index finger
in the direction of a, your middle finger in the direction
of b, and then your thumb points in the direction of n. So let's do that. I'm looking at my hand. It's not an easy thing to do
with your right hand, but your right hand is going to look
something like this. Your index finger will go
in the direction of a. Your middle finger goes
in the direction of b. So that's my middle finger. And then my other two fingers
just do what they need to do. I like to just bend them
out of the way. So they just curl
around my hand. And then what direction
is my thumb in? My thumb-- well, actually I drew
it at the wrong angle. My thumb is actually going
in this direction, right? Into the page. This is the top of my hand. These are like my veins. Or, if I actually drew it
correctly, where you would see your hand from side-- so it
would look like this. You would see your pinky. Your palm and your pinky
would be like that. And your other finger
like this. Your middle finger would go
in the direction of b. Your index finger goes in the
direction of a, and you wouldn't even see your thumb,
because your thumb is pointing straight down. But I think you get the point.
a cross b, this n vector is pointing straight down. It's a unit vector. And this provides
the magnitude. Unit vector just means it
has a magnitude of one. So the magnitudes of the cross
and the dot products seem pretty close. They both have the magnitude
of both vectors there. Dot product, cosine theta. Cross product sine of theta. But then, the huge difference
is that sine of theta has a direction. It is a different
vector that is perpendicular to both of these. Now, let's get the intuition. And if you've watched the videos
on the dot and the cross product, hopefully you
have a little intuition. But I review it because I think
it all fits together when you see them
with each other. First, let's study a,
b cosine of theta. If you watched the dot product
video, cosine of theta, if you took, let's say, b
cosine of theta. What is b cosine of theta? b cosine of theta-- and you
could work it out on your own time-- if you say cosine is
adjacent over hypotenuse, the magnitude of b cosine theta is
actually going to be the magnitude of, if you dropped a
perpendicular-- I'll use a different color here-- if you
dropped a perpendicular here, this length right here--
that's b cosine theta. Let me draw it separately. I don't want to mess up
this picture too much. So if that's b. If that's a-- And that's b. That's a. This is theta. b cosine theta, if you drop a
line perpendicular to a, this is a right angle. b cosine theta, adjacent
over hypotenuse is equal to cosine theta. So it would be the projection
of b going in the same direction as a. So it would be this magnitude. That is b cosine theta. So the magnitude of that vector
right there is the magnitude of b cosine
of theta. So when you're taking the dot
product, at least the example I just did, if you view it as
the magnitude of a times the magnitude of b cosine theta,
you're saying what part of b goes in the same
direction as a? And whatever that magnitude is,
let me just multiply that times the magnitude of a. And I have the dot product. Let's take the pieces that
go the same direction and multiply them. So how much do they
move together? Or do they point together? Or you could view it
the other way. You could view the dot product
as-- and I did this in the dot product video-- you could view
it as a cosine of theta, b. Because it doesn't matter. These are all scalar quantities,
so it doesn't matter what order you take
the multiplication in. And a cosine theta is
the same thing. It's the magnitude of the a
vector that's going in the same direction of b. Or the projection of a onto b. So this vector right here is a
cosine theta; the magnitude of a cosine theta. And they're actually
the same number. If you take how much of b goes
in the direction of a, and multiply that with the magnitude
of a, that gives you the same number as how much of
a goes in the direction of b, and then multiply the
two magnitudes. Now, what is a, b sine theta? a, b, sine theta. Well if this vector right here
is a cosine theta-- and you learned this when you learned
how to take the components of vectors. This vector right here is the
magnitude of a sine theta. You could rewrite this as the
magnitude of a sine theta times the magnitude of b in that
normal vector direction. So if you take a sine theta
times b, you're saying what part of a doesn't go the
same direction as b. What part of a is completely
perpendicular to b-- has nothing to do is b. They share nothing in common. It goes in a completely
different direction. That's a sine theta. And so you take the product of
this with b and then you get a third vector. And it almost says,
how different are these two vectors? And it points in a different
direction. It gives you this-- sometimes
it's called a pseudo vector, because it applies
to some concepts that are pseudo vectors. But the most important of these
concepts is torque, when we talk about the magnetic
field; the force of a magnetic field on electric charge. These are all forces, or these
are all physical phenomena, where what matters isn't the
direction of the force with another vector, it's the
direction of the force perpendicular to
another vector. And so that's where the cross
product comes in useful. Anyway, hopefully, that gave
you a little intuition. And you could have done
it the other way. You could have written
this as b sine theta. And then you would have said
that's the component of b that is perpendicular to a. So b sine theta actually would
have been this vector. Or let me draw it here. That would make more sense. This would be b sine theta. So you could switch orders. You could visualize
it either way. You could say this is the
magnitude of b that is completely perpendicular to a,
multiply the two, and use the right hand rule to get
that normal vector. And we just decided that we're
going to use the right hand rule to have a common
convention. But people could have used the
left hand rule, or they might have used it a different way. It's just a way that we have a
consistent framework, so that when we take the cross product
we all know what direction that normal vector
is pointing in. Anyway. In the next video I'll show you
how to actually compute dot and cross products when
you're given them in their component notation. See you in the next video.