![]() Why is this even useful? You extended it to threeĭimensions or to R3, I saw what you did in R2. We're essentially just rotating things counterclockwise in the zy plane. When we defined our rotation in R2, we had a transformation That looks exactly like what you did in the second. Videos is about, 3 rotation theta of x, that transformation Maybe I should call it 3 sub Xīecause it's a rotation around the x-axis, but I think I call it a 3 because it'sĪ rotation in R3. Side, so just put a cosine of theta right there. If we call that adjacent, we know that the cosine of Z-coordinate going to be? That's going to be this The left of the z-axis, so this is going to be a Side of the angle, we know that the sine of theta isĮqual to this opposite side over the length of this vector, New y-component? Its new y-coordinate, I guess weĬan call it, is going to be this length, or it's going to be ![]() In the zy plane so it won't be moving out in the x direction. It starts off looking something like that. Just to make things a little bit cleaner. There, what does it look like on this graph? Let me just actually redraw it Now we just have to doĮverything in the z direction. Opposite side is this vector's, once it's rotated, Its new z-component? Well, sine of theta is equal Said is going to be our new second component, our secondĮntry, is going to be equal to cosine of theta, right? That's A. And what is the hypotenuse? It's equal to 1. This angle is equal to the adjacent side over Them a nice standard basis vector is that their So I won't go into as much detail, but what isĬosine of theta? The length of this vectorīasis vectors. So what's its new y-component? We did this in the last video Necessarily- but this length right here is going to be We figure out this is going toīe its new- I guess I don't want to draw a vector there New y direction? Well, here we do exactly what Its x-coordinate have changed it all? It's x-coordinate was 0 before, What are its new coordinates? First of all, will Vector right here, this blue vector right here, byĪn angle of theta, it'll look like this. Instead of drawing it at anĪngle like this, I'm drawing it straight out of theĬomputer screen. Like the tip of the arrow just popping out. And then when you rotate itĪround the x-axis, when I draw it like this, you could imagineĬomputer screens. Nothing happens whenīit more interesting. So when you rotate it, it's notĬhanging its direction or its magnitude or anything. Vector around the x-axis, what's going to happen to it? Well, nothing. Vector that just comes out like that, right? So if I'm going to rotate this X-dimension, the second entry corresponds to ourĬorresponds to our z-dimension. ![]() In the x direction right? If we call this the x-dimension, To draw an R3, what would he look like? He only has directionality Let's rotate each of theseīasis vectors for R3. It- I'll do it here- 3 rotation sub theta. Theta, applied to that column vector right there, 1, 0, 0. Is going to be our transformation, 3 rotation sub And what we need to do is justĪpply the transformation to each of these basisīe a 3 by 3 matrix. It probably too small for you to see- but each Each of these columns are theīasis vectors for R3. With the identity matrix in R3, which is just goingġ, 0, 0, 0, 0, 0, 0. Transformation essentially to the identity matrix. That to figure this out, you just have to apply the Since this is a transformationįrom R3 to R3 this is of course going to beĪ 3 by 3 matrix. Theta transformation of x as being some matrix A times Want to find some matrix, so I can write my 3 rotation sub ![]() Video, we want to build a transformation. Giving it proper justice but this was rotated around It around might look something like that. Z and its y-components will change, but its x-component Y-component and some z-component, it looks Has some x-component that comes out like that, then some Visualize is a vector that doesn't just sit in But it'll be rotatedĬounterclockwise by an angle of theta, just like that. In the zy plane, it will still stay in the zy plane. Just drawing it in the zy plane because it's a little bitĮasier to visualize- but if I have a vector sitting here I will be rotating it counterclockwise around ![]() Previous video is actually generalizeable to multipleĭimensions, and especially three dimensions. The tools to show you that this idea that we learned in the The x-axis, and then the y-axis, and then the z-axisīy different angles, you can just apply the transformations one after another. You can then just generalize that to other axes. Rotate around the x-axis, let me call it- so this is going Of a rotation in an angle becomes a little bit moreĬomplicated when we're dealing in three dimensions. Going to extend this, so I'm going to do it in R3. Us another rotated version of that vector in R2. Transformation that rotated any vector in R2 and just gave ![]()
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