Vector Equation of a Tangent Line on a Space Curve by @mes on 3Speak
In this video, I go through obtaining the vector equation of a tangent line to a space curve, by replacing the parallel vector with the tangent vector from the general vector equation of a line. Th...
Tags: #math #calculus #science #education #tangent #scifimultiverse #vectors #threespeak #chessbrothers
The solution to question 3, and that's a tangent vector. So how do you find the tangent vector to a smooth curve at a point? And then we're asked, how do you find the tangent line, and then the unit tangent vector? Alright, let's take a look at the solution. So let's see be a smooth curve with position vector R of T. Then basically, the tangent vector is going to be R prime of T. It's a basic derivative of this one. And then the unit tangent vector is basically just a tangent vector, but it has to be magnitude 1 or length 1. So the unit tangent vector, this capital T of T, a small case T, equals to just a derivative. So the unit tangent vector, divided by the length of the unit tangent vector. So just so that this entire vector, we normalize it so that the length is going to be one. So then the magnitude of this. That's all it is. Alright, so now going further, so I just quickly recall from my earlier video. Again, there's link to the video notes, and then you'll see all these thumbnails, scroll down to the bottom, and you'll see that. And then, you know, recall my earlier video on the definition of derivative of a vector function, and basically you take the secan vector, divided by H and then take the limit as H apart 0. And basically, yeah, we just get our derivative there as a tangent vector. And then the unit tangent vector is this right here. So R prime divided by, yeah, R prime of T divided by the magnitude of R prime of T. All right, so now recall also my earlier video for the vector equation of a line. And that's because we're going to go look at how to find the tangent line. All right, so recall the vector equation of a line. So we have a line like this, L. and then this is going to be, this R vector, it's going to be denoting the line, so at all points of it, and then it equals to basically just the initial line, initial vector, R0, so at a point A there, or actually at a point, just at a point not there, or whatever, X not, or T, yeah, so then you would plus T times by a parallel vector, so this is the parallel vector to this one here. All right, and then you add those up, and then basically you're going to have this, basically, these are this plus A, where A is equal to TV. So that's the equation of a line, so basically just using vector addition, so add them up and equals to this vector. So this plus this equals to this one here. Alright, so now that brings us to how to solve the vector equation of a tangent line at t equals A. So let's just draw this out to make it a bit nicer. So I'll just grasp it a bit. Yeah, just because I actually haven't, I didn't do this in the earlier videos. So if we have this, like, there's the origin, this is the Z, this is the Y, this is the X. Or let's say we had a curve like this. Here, I just fixed that up. So let's say this is C, that's our curve. And now let's say we have our R of T. Yeah, let's say this is R of, this is our R function, but this is at the value A. Yeah, this is I'm just going to put this as A here. That's just the one we want. And let's say the tangent vector is going to be parallel to this, I mean, it's going to be tangent to this curve. So that's going to be R prime of A like that. All right. And then the line, yeah, I'm just going to draw this in dash lines, actually, dot, dot, dot, dot. This is going to be our line all the way across. and that's going to be our line L. Alright, so this is what we have, and we basically have to make it match up to this, and notice how, again, this is parallel to this right there, A and V, and so this one's actually, this one, what it's actually is, it's going to be, we're going to add a parameter, I'm going to call this U, so U parameter times it by that, and this is going to be, let's say, our A. This is our, well, they have, this A vector is that one, so we'll have a different vector. Yeah, I'll just call this B vector. So that's the B vector. So this equals to parameter, just a U there, and this one, because this one's T is already taken for just the normal vector. So we're going to call that U, so B vector is equal to U. I mean, u multiplied by that. So it's not a dot part because it's just a scalar. So anyways, and then the next setup is this is going to be parallel to this. So the parallel vector, I'll draw it actually like this. This is going to be, now remove this, this is going to be parallel to this. So that's our parallel vector. This is going to be R prime of A like that. And then, what I'll do is I'm going to write this right here. This is going to be, so this vector, that's going to be our R of A vector. So that's going to be our initial vector. And yeah, so that's going to be this part right here. So it's R0. That's going to be the R0. and then the next setup is we're going to need a vector all the way over there, so that we're adding the two together. So I'll make one like this. All right, so that's going to be a vector like that. All right, so I moved it all the way over there. I'm going to call this now S. So this is going to be S, a vector at, and the parameter's U. So basically, when a parameter is shorter, it goes down here, or negative, if it's longer, it goes over here, et cetera. Basically, all it does is extends that parallel vector to it, and that parallel vector is the tangent vector. All right, so what that means is, well, this becomes, this line now is going to be S of U. Yeah, so S of U vector like this. This is going to be equal to the initial vector is going to be R of A, so at the, when U equals A, and then plus the parameter U, so again, same thing as instead of dealing with this, T or R0, we're switching it up, this is going to be RA, and it's going to be U, just change up the variable, and this is going to be now times it by the parallel vector V, which is now the derivative of the vector R of A. like that so we can box this in so that's yeah that's pretty fascinating stuff