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 and 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 data 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 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 you 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 have 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 R cosine or the inverse cosine there are the equivalent functions and then 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 well that 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 time's the length of B times the sine of the angle between them times the sine 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 a diversion there just to kind of show you how to solve for theta because I realize 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 time's 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 my little line the middle 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 is the length of that line right there so that is the length of B so what is let me say let me let me rewrite this thing up here let me write it as B the length of B times the length I want to be careful I don't want to do the dot there because you 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 time's the cosine of theta let's get out our basic trigonometry tools sohcahtoa so Chi Toa cosine 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 then what is the cosine of theta it's equal to it's equal to this let me do it in 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 now I want to do a little bit bigger it's equal to the adjacent side over the hypotenuse right so let me write this down so cosine of theta 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 that's not a night 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 I get the length of vector a times the cosine of theta is equal to the adjacent side I'll do that in magenta is equal to the adjacent side so this expression right here which was just a dot B can be rewritten as I just told you that the met 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 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 right 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 if the word projection helps you just you think of it 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 if they 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 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'll 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 look like that vector a look like that and my vector B look like that well now the adjacent the way I define it here if I had to drop a if I had to make a right triangle like that the adjacent sides very small so your 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 could do it the other way you could draw a a 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 or arbitrary so the takeaway is the fact that the fact that a dot B is equal to the lengths of each of those times the cosine of theta two it says that the dot product so the dot product tells me how much are my vector is 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 same direction right you could view this adjacent side you could view this adjacent side here as the part of a that's going in the direction of B right 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 - when - things are orthogonal or when they're perpendicular when a dot B is equal to zero 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 you know like if I had to draw a right triangle would come straight down and if I were to say the projection of a and I haven't drawn that if I put a light shining down from above and I'd say what what's the shadow of a onto B you'd get nothing you'd get zero this arrow has no width even though I've drawn it to have some width it has no width so you would have a zero down here the part of a that goes in the same direction is B no part of this vector goes in the same direction as this vector so you're going to have this zero kind of adjacent side times B so you're going to get something that's zero 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 shows you is equal to the length of a time's the length of B times the sine of the angle between them so let me do the same example let me draw two my two vectors that's my vector a this is my vector B and now sine sohcahtoa so sine of theta let me write that sine of theta sohcahtoa is equal to opposite over the hypotenuse so if I were to draw a little right triangle here so if I were to drop a perpendicular right there and this is theta what is the sine of theta equal to in this context the sine of theta is equal to what it's equal to this side over here let me call that just the opposite 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 iPod noose 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 sine of theta is equal to the opposite side so if we 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 it with the dot product as well this is equal to be the length of vector B times the length of vector a sine 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 our the length of vector B times the part of a that's going perpendicular to B right this part of this this opposite side is the part of a that's going perpendicular to B so they're 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 and 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 so you're saying you're kind of 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 if you take two right triangle so that that's a and that's or if you take two vectors that are perpendicular to each other the length of a cross B a cross B is he going to be equal to if we just use this formula right there the length of a time's the length of B and what's the sine of 90 degrees it's one so in this case you kind of have maximized the the length of your cross product this is as high as it can go because sine of theta its maximum value sine of theta is always less than or equal to one so this is as good as you're ever going to get this is the highest possible value and you have perfect perfectly perpendicular vectors now when is actually just to kind of go back to make the same point here when you get the maximum value for your cosine of for your dot product well it's when your two vectors are collinear if I have one vector 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 zero 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 the cosine of the angle between them that's cosine of that angle is zero or the cosine of the angle is zero so the cosine that is one so when you have two and two 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 when and you know just to make the analogy clear when they're perpendicular to each other you've minimized or at least the magnitude of your dot but 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 each other similarly if you were to take 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 don't want to draw them on top of each other but I think you get the idea let's say vector B is like that vector B then the side that then theta is 0 you can't even see it they've it's been squeezed out I've just brought these new things on top of each other and then the cross-product in this situation a cross B is equal to well it's the mat the length of both of these things times the sine of theta sine of 0 is 0 so it's just zero so two collinear vectors they're the magnitude of the cross product is zero 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 it's going to be the length of a time's 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 zero anyway I wanted to make all of this clear because you know sometimes you can 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 the cross product tends to give people more trouble so if I have that's my a and that's my B what if I wanted to figure out the area of this parallelogram so if I draw it with a draw if I were to shift a and have that there and if I were to shift B and or 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 I would drop a perpendicular right there I would drop a perpendicular right there if this is perpendicular and this length is H let me 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 times my height but what is my height let me just draw a little theta there let me do a green data it's more visible so theta so we know already that the sine of this theta is equal to the opposite over the hypotenuse so it's equal to the height over the hypotenuse 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 get the height is equal to the length of vector a times the sine 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 sine theta well this is just the length of the cross product of the two vectors a a cross B this is the same thing I mean you can rearrange the a and the B so that should 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 the 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 parallel 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 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