Vector dot and cross products
Dot and cross product comparison/intuition
We've known for several videos now that the dot product of two nonzero vectors, a and b, is equal to the length of vector a times the length of vector b times the cosine of the angle between them. Let me draw a and b just to make it clear. If that's my vector a and that's my vector b right there, the angle between them is this angle. And we defined it in this way. And actually, if you ever want to solve-- if you have two vectors and you want to solve for that angle, and I've never done this before explicitly. And I thought, well, I might as well do it right now. You could just solve for your theta. So it would be a dot b divided by the lengths of your two vectors multiplied by each other is equal to the cosine of theta. And then to solve for theta you would have to take the inverse cosine of both sides, or the arc cosine of both sides, and you would get theta is equal to arc cosine of a dot b over the magnitudes or the lengths of the products of, or the lengths of each of those vectors, multiplied by each other. So if I give you two arbitrary vectors-- and the neat thing about it is, this might be pretty straightforward. If I just drew something in two dimensions right here, you could just take your protractor out and measure this angle. But if a and b each have a hundred components, it becomes hard to visualize the idea of an angle between the two vectors. But you don't need to visualize them anymore because you just have to calculate this thing right here. You just have to calculate this value right there. And then go to your calculator and then type in arc cosine, or the inverse cosine that are the equivalent functions, and it'll give you an angle. And that, by definition, is the angle between those two vectors, which is a very neat concept. And then you can start addressing issues of perpendicularity and whatever else. This was a bit of a tangent. But the other outcome that I painstakingly proved to you in the previous video was that the length of the cross product of two vectors is equal to-- it's a very similar expression. It's equal to the product of the two vectors' lengths, so the length of a times the length of b times the sine of the angle between them. Times the sign of the angle between them. So it's the same angle. So what I want to do is take these two ideas and this was a bit of a diversion there just to kind of show you how to solve for theta because I realized I've never done that for you before. But what I want to do is I want to take this expression up here and this expression up here and see if we can develop an intuition, at least in R3 because right now we've only defined our cross product. Or the cross product of two vectors is only defined in R3. Let's take these two ideas in R3 and see if we can develop an intuition. And I've done a very similar video in the physics playlist where I compare the dot product to the cross product. Now, if I'm talking about-- let me redraw my vectors. So the length of a-- so let me draw a. b, I want to do it bigger than that. So let me do it like that. So that is my vector b. So this is b. That is a. What is the length of a times the length of b times the cosine of the angle? So let me do that right there. So this is the angle. So the length of a if I were to draw these vectors is this length right here. This is the length of a. It's this distance right here, the way I've drawn this vector. So this is, literally, the length of vector a. And I'm doing it in R3 or maybe a version of it that I can fit onto my little blackboard right here. So it'll just be the length of this line right there. And then the length of b is the length of that line right there. So that is the length of b. Let me rewrite this thing up here. Let me write it as b, the length of b times the length-- and I want to be careful. I don't want to do the dot there because you'll think it's a dot product. Times a cosine of theta. All I did is I rearranged this thing here. It's the same thing as a dot b. Well what is a times the cosine of theta? Let's get out our basic trigonometry tools-- SOH CAH TOA. Cosine of theta is equal to adjacent over hypotenuse. So if I drop, if I create a right triangle here, and let me introduce some new colors just to ease the monotony. If I drop a right triangle right here and I create a right triangle right there, and this is theta, than what is the cosine of theta? It's equal to this. Let me do it another color. It's equal to this, the adjacent side. It's equal to this little magenta thing. Not all of b, just this part that goes up to my right angle. That's my adjacent. I want to do it a little bit bigger. It's equal to the adjacent side over the hypotenuse. So let me write this down. So cosine of theta is equal to this little adjacent side. I'm just going to write it like that. Is equal to this adjacent side over the hypotenuse. But what is the hypotenuse? It is the length of vector a. It's this. That's my hypotenuse right there. So my hypotenuse is the length of vector a. And so if I multiply both sides by the length of vector a I get the length of vector a times the cosine of theta is equal to the adjacent side. I'll do that in magenta. So this expression right here, which was just a dot b can be rewritten as-- I just told you that the length of vector a times cosine of theta is equal to this little magenta adjacent side. So this is equal to the adjacent side. So you can view a dot b as being equal to the length of vector b-- that length-- times that adjacent side. And you're saying, Sal, what does that do for me? Well what it tells you is you're multiplying essentially, the length of vector b times the amount of vector a that's going in the same direction as vector b. You can kind of view this as the shadow of vector a. And I'll talk about projections in the future. And I'll more formally define them, but if the word projection helps you, just think of that word. If you have a light that shines down from above, this adjacent side is kind of like the shadow of a onto vector b. And you can imagine, if these two vectors-- if our two vectors looked more like this, if they were really going in the same direction. Let's say that's vector a and that's vector b, then the adjacent side that I care about is going to-- they're going to have a lot more in common. The part of a that is going in the same direction of b will be a lot larger. So this will have a larger dot product. Because the dot product is essentially saying, how much of those vectors are going in the same direction? But it's just a number, so it will just be this adjacent side times the length of b. And what if I had vectors that are pretty perpendicular to each other? So what if I had two vectors that were like this? What if my vector a looked like that and my vector b looked like that? Well now the adjacent, the way I define it here, if I had to make a right triangle like that, the adjacent side's very small. So you're dot product, even though a is still a reasonably large vector, is now much smaller because a and b have very little commonality in the same direction. And you can do it the other way. You could draw this down like that and you could do the adjacent the other way, but it doesn't matter because these a's and b's are arbitrary. So the take away is the fact that a dot b is equal to the lengths of each of those times the cosine of theta. To me it says that the dot product tells me how much are my vectors moving together? Or the product of the part of the vectors that are moving together. Product of the lengths of the vectors that are moving together or in the same direction. You could view this adjacent side here as the part of a that's going in the direction of b. That's the part of a that's going in the direction of b. So you're multiplying that times b itself. So that's what the dot product is. How much are two things going in the same direction. And notice, when two things are orthogonal or when they're perpendicular-- when a dot b is equal to 0, we say they're perpendicular. And that makes complete sense based on this kind of intuition of what the dot product is doing. Because that means that they're perfectly perpendicular. So that's b and that's a. And so the adjacent part of a, if I had to draw a right trianlge, it would come straight down. And if I were to say the projection of a and I haven't draw that. Or if I put a light shining down from above and I'd say what's the shadow of a onto b? You'd get nothing. You'd get 0. This arrow has no width, even though I've drawn it to have some width. It has no width. So you would have a 0 down here. The part of a that goes in the same direction as b. No part of this vector goes in the same direction as this vector. So you're going to have this 0 kind of adjacent side times b, so you're going to get something that's 0. So hopefully that makes a little sense. Now let's think about the cross product. The cross product tells us well, the length of a cross b, I painstakingly showed, you is equal to the length of a times the length of b times the sin of the angle between them. So let me do the same example. Let me draw my two vectors. That's my vector a and this is my vector b. And now sin-- SOH CAH TOA. So sin of theta, let me write that. Sin of theta-- SOH CAH TOA-- is equal to opposite over the hypotenuse. So if I were to draw a little right triangle here, so if I were to draw a perpendicular right there, this is theta. What is the sin of theta equal to in this context? The sin of theta is equal to what? It's equal to this side over here. Let me call that just the opposite. It's equal to the opposite side over the hypotenuse. So the hypotenuse is the length of this vector a right there. It's the length of this vector a. So the hypotenuse is the length over my vector a. So if I multiply both sides of this by my length of vector a, I get the length of vector a times the sin of theta is equal to the opposite side. So if we rearrange this a little bit, I can rewrite this as equal to-- I'm just going to swap them. I have to do the dot product as well. This is equal to b, the length of vector b, times the length of vector a sin of theta. Well this thing is just the opposite side as I've defined it right here. So this right here is just the opposite side, this side right there. So when we're taking the cross product, we're essentially multiplying the length of vector b times the part of a that's going perpendicular to b. This opposite side is the part of a that's going perpendicular to b. So they're kind of opposite ideas. The dot product, you're multiplying the part of a that's going in the same direction as b with b. While when you're taking the cross product, you're multiplying the part of a that's going in the perpendicular direction to b with the length of b. It's a measure, especially when you take the length of this, it's a measure of how perpendicular these two guys are. And this is, it's a measure of how much do they move in the same direction? And let's just look at a couple of examples. So if you take two right triangles. So if that's a and that's-- or if you take two vectors that are perpendicular to each other, the length of a cross b is going to be equal to-- if we just use this formula right there-- the length of a times the length of b. And what's the sin of 90 degrees? It's 1. So in this case you kind of have maximized the length of your cross product. This is as high as it can go. Because sin of theta, it's a maximum value. Sin of theta is always less than or equal to 1. So this is as good as you're ever going to get. This is the highest possible value when you have perfectly perpendicular vectors. Now, when is-- actually, just to kind of go back to make the same point here. When do you get the maximum value for your cosine of-- for your dot product? Well, it's when your two vectors are collinear. If my vector a looks like that and my vector b is essentially another vector that's going in the same direction, then theta is 0. There's no angle between them. And then you have a dot b is equal to the magnitude or the length of vector a times the length of vector b times the cosine of the angle between them. The cosine of the angle between them, the cosine of that angle is 0. Or the angle is 0, so the cosine of that is 1. So when you have two vectors that go exactly in the same direction or they're collinear, you kind of maximize your dot product. You maximize your cross product when they're perfectly perpendicular to each other. And just to make the analogy clear, when they're perpendicular to each other you've minimized-- or at least the magnitude of your dot product. You can get negative dot products, but the absolute size of your dot product, the absolute value of your dot product is minimized when they're perpendicular to each other. Similarly, if you were to take two vectors that are collinear and they're moving in the same direction, so if that's vector a, and then I have vector b that just is another vector that I want to draw them on top of each other. But I think you get the idea. Let's say vector b is like that. Then theta is 0. You can't even see it. It's been squeezed out. I've just brought these two things on top of each other. And then the cross product in this situation, a cross b is equal to-- well, the length of both of these things times the sin of theta. Sin of 0 is 0. So it's just 0. So two collinear vectors, the magnitude of their cross product is 0. But the magnitude of their dot product, the a dot b, is going to be maximized. It's going to be as high as you can get. It's going to be the length of a times the length of b. Now the opposite scenario is right here. When they're perpendicular to each other, the cross product is maximized because it's measuring on how much of the vectors-- how much of the perpendicular part of a is-- multiplying that times the length of b. And then when you have two orthogonal vectors, your dot product is minimized, or the absolute value of your dot product. So a dot b in this case, is equal to 0. Anyway, I wanted to make all of this clear because sometimes you kind of get into the formulas and the definitions and you lose the intuition about what are all of these ideas really for? And actually, before I move on, let me just make another kind of idea about what the cross product can be interpreted as. Because a cross product tends to give people more trouble. That's my a and that's my b. What if I wanted to figure out the area of this parallelogram? If I were to shift a and have that there and if I were to shift b and draw a line parallel to b, and if I wanted to figure out the area of this parallelogram right there, how would I do it just using regular geometry? Well I would drop a perpendicular right there. This is perpendicular and this length is h for height. Then the area of this, the area of the parallelogram is just equal to the length of my base, which is just the length of vector b times my height. But what is my height? Let me just draw a little theta there. Let me do a green theta, it's more visible. So theta. So we know already that the sin of this theta is equal to the opposite over the hypotenuse. So it's equal to the height over the hypotenuse. The hypotenuse is just the length of vector a. So it's just the length of vector a. Or we could just solve for height and we'd get the height is equal to the length of vector a times the sin of theta. So I can rewrite this here. I can replace it with that and I get the area of this parallelogram is equal to the length of vector b times the length of vector a sin theta. Well this is just the length of the cross product of the two vectors, a cross b. This is the same thing. I mean you can rearrange the a and the b. So we now have another way of thinking about what the cross product is. The cross product of two vectors, or at least the magnitude or the length of the cross product of two vectors-- obviously, the cross product you're going to get a third vector. But the length of that third vector is equal to the area of the parallelogram that's defined or that's kind of-- that you can create from those two vectors. Anyway, hopefully you found this a little bit intuitive and it'll give you a little bit more of kind of a sense of what the dot product and cross product are all about.