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# Dot vs. cross product

## Video transcript

let's do a little compare and contrast between the dot product and the cross product so let me just make two vectors just let me visually draw them and maybe if we have time we'll actually figure out no that's too thin well actually figure out some dot and cross products with real vectors and let me make another one I always make a relatively acute angle maybe I'll make an up well I always do the acute angle okay let's call the first one I'll just label them take forever if I keep switching colors that's the angle between them okay 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 does a dot be that 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 you end up with just a number and that is equal to the magnitude of a time's the magnitude of B times cosine of the angle between them fair enough 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's actually equals the 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 diverges this times the sine of the angle between them the sine of the angle between them and this is where it really diverges right this when we took the dot product we just ended up with a number this is just a number this there's no direction here this is just a scalar quantity but the cross product we take the magnitude of a time's the magnitude of B times the sine of the angle between them and that provides the magnitude but it also has a direction and the direction is provided by this normal vector which I'll just it's a unit vector unit vector gets that little hat on it it's a unit vector in what direction is it well that's defined by the right-hand rule it's a it so this is a vector it's it's perpendicular so perpendicular perpendicular to both a and B and B and then well you might say okay 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 learn about the cross product I said you know there's two ways of you know to show a vector popping out of the screen and 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 a the back of an hour 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 well that's where you take your let 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 porns in the in the direction of n so let's do that I'm looking at my hand and it looks like it's actually not an easy thing to do with your right hand but your right hand is going to look something like this your index finger is going to well let me actually do it in the direction of a so 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 I'm trying my best to draw so they just curl around my hand and then what direction is my thumb and my thumb well actually I drew it in the wrong angle I drew it my thumb is actually going in this direction right into the page this is the top of my hand these are like my veins right or if I actually drew it correctly where it would actually you would see your to your hand from the side so it would look like this you would see your pinky you'd see like put your palm and your pinky would be like that then 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 actually 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 product seem pretty close they both have the magnitudes of both vectors their 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'll review it because I think it all fits together when you see them with each other let me do some erase no that's not what I wanted to do no that's not what I want to do I could erase this way okay so first let's study what a B cosine of theta so if you watch the dot product video cosine of theta when you take if you took let's say B cosine of theta what is B cosine of theta B cosine of theta and you can work it out on your own time if you say cosine is adjacent over hypotenuse B cosine of theta is actually going the magnitude of B cosine theta is actually going to be the magnitude if you dropped a perpendicular I'll do it in 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 now I'll get the same a no I was going to use a line tool if that's a and now the rest of it I'll just doing one color to save time if that's B that's a this is theta B cosine theta is if you drop a pop for a line perpendicular to a this is right angle B cosine theta adjacent over hypotenuse is equal to cosine theta so it'd 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 mat the magnitude of that vector right there is B the magnitude of B cosine of theta so when you're taking the dot product at least the example I just did it with you if you say if you view it as the magnitude of a time's 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 so how let's take the pieces that go in the same direction and multiply them so how much do they move together or do they point to get 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 right because it doesn't matter these are all scalar quantity so it doesn't matter what order you 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 as 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 take and take the 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 learn how to take the components of vectors this vector right here is the magnitude of a sine theta right you could rewrite this as a 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 in the same direction as B what part of a is completely perpendicular to B has nothing to do with B they share nothing in common it goes 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 kind of 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 the force of a magnetic field on an electric charge these are all forces or these are all physical phenomenon 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 oh well that's the component of B that is perpendicular to a so B sine theta actually would have been this vector right or let me draw it here that'd make more sense this is this would be B sine theta so you could you know you could switch orders you could visualize it either way you could say this is the magnitude of B this completely perpendicular to a you multiply the two and then use the right hand rule to get that normal vector and that right hand rule there's nothing we just decided you know 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 2 notation see in the next video