 Now you'll hear a lot about the normal the tangent normal coordinate system Now it plays on the strengths of what we've just done here this curvature and all these things actually Go together and you can use them together. So let's go for it First things first. I just want you to consider a circle Consider a circle Circular motion goes round and round and round It is the only kind of movement that oh, that's what that's our unhorrible circle. Anyway It's the only movement that you can find in which the magnitude of the position vector never changes It just as it goes that the magnitude this radius there the stays constant all the time So it's the only type of movement that you can really think of in which in this initial difference frame where this The magnitude of this position vector never changes and as such it's first derivative The velocity vector at any point is always going to be perpendicular To that the position vector that the velocity vector always going to be perpendicular and For us to look at a body coordinate system now imagine we've all seen these action cameras mounted on a helmet on a motorbike or bicycle and It's as if the helmet of the person is absolutely still that's a body coordinate system and this whole world is moving Around now imagine if you are in a particle Point moving along this trajectory as far as you are concerned you have your own coordinate system And the whole world is moving around you and that is a coordinate system that is different from this World view this Cartesian coordinate system. That's a body coordinate system a coordinate system that is going to remain static for from the point of view of Someone or something on that body on that particle that moves along so for that we have this Tangent normal tangent normal Bynormal section some people put the B in and I'll show you what the bynormal unit vector is That is a coordinate system that is fixed on this moving particle and it's fast It's concerned the whole world is moving around it now. We already know this tangent normal vector The tangent normal vector is just the norm of the velocity vector So at any point that I can that I have a position vector at any point that I have a position vector I take its first derivative and I have the velocity vector in other words I have our prime of t That is the first derivative there and if I just expressed its magnitude divided by its magnitude Prime of t So I have a unit vector if I do that its length becomes one And all it now shows me is the direction and that is the tangent normal vector We've seen that the tangent normal vector Now I Want to construct as I have with a Cartesian coordinate system? I have here this it is perpendicular to each other So I want to construct the same sort of thing there. So at this point. I want if that's my T of t there. I want something that is perpendicular to that at any time and I'm going to call that the principal normal Principal normal vector. It is perpendicular to that and At any point I sort of instantaneously want to see that as This sort of scenario if I have a position vector and I take its first derivative in this sort of scenario Which is happening instantaneously at every kind of point you can see it is something like that I want this kind of scenario so that this is my t and this is my n And what do I do? I take the derivative of that So we're going to have this principal normal unit vector principal normal unit vector that is going to be for us t prime of t over its Magnitude so again, it's going to have a length of one So what it basically tells us it's this it's a prime there It's the it goes together with this. It's the derivative of that And divided by its magnet. It's magnitude just to make a unit vector again But it shows us for this side type of scenario now Don't mistake this for this this particle is not moving in a circle But as it stands right at the spot to view it as instantaneously as looking like something like this and if you were to do this You can you can do we can do a problem or you can do problems where you can just take the the dot product Of these two vectors is going to end up being zero in other words. They are perpendicular to each other Now you can well imagine in two-dimensional space that this Normal principal normal tangent vector can point in this direction or it can point in this direction And what it tends to do is it always points in the direction in which This is changing if the slope is changing in this direction at the moment Okay, so it's concave looking from this side. It's going to point in that direction Over here suddenly it's going to swap over looking in that direction in three-dimensional space. It's actually even More difficult because it can not only point in one of two directions But it can point anyway in a circle And there is something called the oscillate oscillating circle in the in the circle in the plane We need to be too concerned about these These things now But what the point of it all is that this usually points in the direction in which The curvature is going at the moment. Okay, it's easy enough to see And then we have this binormal unit vector b That you just have to know about And that is just the cross product That is just the cross product of these two vectors Just the cross product of those two vectors. So nothing here is too difficult too sinister Now we are going on to another form of The another derivation so that you have another equation to play with to determine curvature And for that we're going to make use of this tangent normal Or body coordinate system That stays with the particle wherever wherever it goes