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Current time:0:00Total duration:10:53

Video transcript

let's see if we get a little bit more practice and intuition of what cross products are all about so in the last example we took a cross B let's see what happens when we take B cross a so let me let me erase some of this let me erase I don't want to erase all of it because it might be useful to give us an intuition to compare so let's say I'm going to keep that I'm going to keep that actually I can erase this I think okay let me see okay so these the things I have drawn here this was a cross B let me cordon it off so that you don't get confused so that was me using the right hand rule when I try to do a cross B and then we saw that the unit the the magnitude of this was 25 and n the the direction pointed into pointed downwards or when I drew it here it would point into the page so let's see what happens with B cross a so I just I'm just switching the order B cross a well the magnitude is going to be the same thing right because I'm still going to take the magnitude of B times the magnitude of a times the sine of the angle between them which was almost PI over 6 radians and then times some unit vector n but this is going to be the same when I multiply scalar quantities it doesn't matter what order I multiply them in right so this is still going to be 25 whatever my units might have been times some vector N and we still know that that vector n it has to be perpendicular to both a and B and now we have to figure out well is it in being perpendicular can either kind of point into the page here or it could pop out of the page or point out of the page so which one is it and then we take our right hand out and we try it again so what we do is we take a right hand and let's see if I can I'm actually drawing my I'm using my right hand right now although you can't see it just to make sure I draw the right thing so in this example if I take my right hand I take the index finger in the direction of B I take my middle finger in the direction of a so my middle figure is going to look something like that right and then I have two leftover fingers there then the thumb goes in the direction of the cross product right because your thumb has a right angle right there that's the right angle of your thumb so in this example if this is that's the direction of a this is the direction of B and we're doing B cross a that's why B gets your index finger the index finger gets the first term your middle finger gets the second term and the thumb gets the direction of the cross product so in this example the direction of the cross product is upwards or in this when we're drawing in two dimensions right here the cross product would actually pop out of the page for B cross a so I'll write draw it over B the circle with the dot or if I were to draw it analogous to this so this is this right here that was a cross B and then B cross a is the exact same magnitude but it goes into the other direction that's B cross a it just flips the opposite direction that's why you have to use your right hand because you might know that oh something's going to pop in or out of the page etc etc but you need to know your right hand too to know whether it goes in or out of the page anyway let's see if we can get a little bit more intuition of what this is all about because this is all about intuition and frankly I'll tell you the cross product comes into use and to a lot of concepts that frankly we don't have a lot of real-life intuition with you know electrons flying through a magnetic field or magnetic field through a coil and a lot of things that you know in our everyday life experience maybe if we if we were metal filings living in a magnetic field our entire well we do live in a magnetic field in a strong magnetic field maybe we would get an intuition but it's hard to have as deep of an intuition as we do for say falling objects or friction or forces or fluid dynamics even because we've all played with water but anyway let's get a little bit more intuition and let's let's think about why is there that sine of theta you know want to just multiply the magnitudes times each other and use the right hand rule and figure out a direction what is that sine of theta all about and I think I need to I need to clear this up a little bit just see you can beep this could be useful so why is that sine of theta there let's let me draw a redraw some vectors I'll draw them a little fatter so let's say that's a that's a this is B B doesn't always have to be longer than a I just key so this is a and this is B now we could think of a little bit we could say well this is the same thing as a sine theta times B or we could say this is B sine theta times a hope I'm not confusing is you can interpret this as this could also be written as because these are just magnitudes right so it doesn't matter what order you multiply them in you could say this is a sine theta times the magnitude of B all of that in the direction of the normal vector or you could put the sine theta the other way but let's think about what this would mean a sine theta if this is Theta what is a sine theta sine is opposite over hypotenuse right so opposite over hypotenuse so this would be the vector the magnitude of a if I were to drop let me draw something let me draw a line here make it a real line let me draw a line there so I have a right angle a right angle right there so what's a sine theta and this is the opposite side so a sine theta is a and with sine of theta is opposite over hypotenuse the hypotenuse is the magnitude of a right so sine of theta is equal to this side which I call Oh for opposite us opposite over the magnitude of a so it's opposite over the magnitude of a so this term a sine theta is actually just the magnitude of this line right here another way you could let me let me redraw it I could it doesn't matter where the vector start from all you care about is this magnitude and direction so you can shift vectors around so this vector right here you could call it this opposite vector that's the same thing as this vector that's the same thing as this I just shifted up away and so another way to think about it is it is the component of vector a right we're used to taking a vector and splitting it up into you know X and y components but now we're taking a vector a and we're splitting it up into you could think of it as a component that's parallel to vector B and a component that is perpendicular to vector B so a sine theta is the magnitude of the the component of vector a that is perpendicular to B so what when you're taking the cross product of two numbers you're kind of saying well I don't care about the entire magnitude of say vector a in this example I care about the magnitude of vector a that is perpendicular to vector B and those are the two numbers that I want to multiply and then give it that direction as specified by the right hand rule and I'll show you some applications this is especially important well we will use it in torque and we'll also use it in magnetic fields but but it's it's important in both of those applications to figure out the components of the vector that are perpendicular to either a force or radius in question so that's why this this cross product has the sine theta because we're taking so in this if you view it as magnitude of a sine theta times B this is kind of saying this is the magnitude of the come of the component of a perpendicular to B or you can interpret it the other way you can interpret it as a times b sine-theta alright if put a parentheses here and then you could view it the other way you could say well b sine-theta is the component is the component of b that is perpendicular to a let me draw that just to hit the point home so that's my a that's my B so B is how well this is a this is B so B has some component of it that is perpendicular to a and that is going to look something like well I kind of run out of space let me draw it here if that's a that's B the component of B that is perpendicular de is going to look like this I'll do it in a different color it's going to be perpendicular to a and it's going to go that far right and then you can go back to sohcahtoa and you could prove to yourself that the magnitude of this vector is B sine theta so that is where the sine theta comes from it make sure that we're not just multiplying the vectors and make sure we're multiplying the components of the vectors that are perpendicular to each other to get a third vector that is perpendicular to both of them and then you know the people who invented the the cross products said well it's still ambiguous because it doesn't tell us there's always two vectors as perpendicular to these two one goes in one goes out they're in opposite directions and that's where the right-hand rule comes in they'll say okay well we'll just agree we're just going to say a convention that you use your right hand point it like a gun make all your fingers perpendicular and then you know whether what direction that vector points in anyway hopefully you're not confused now I want you to watch the next video and this is actually gonna be some physics on electricity magnetism and torque and that's essentially the applications of the cross product and it'll give you a little bit more intuition of how to use it see you soon