![rotation rules geometry clockwise rotation rules geometry clockwise](https://www.mathematics-monster.com/images6/rotation_clockwise_and_counter-clockwise.jpg)
To each of these sides? Well, this side right over here, if I rotate it by negative 90 degrees it is going to end up, Rotating this right triangle, this magenta one that I just constructed, by negative 90 degrees,īy negative 90 degrees, or 90 degrees in the clockwise direction. All right, so I have this right triangle. So, it's gonna go, it's gonna be, like, it's gonna look like that, and then it's gonna look like, whoops, that wasn't to press my, Of the right triangles is gonna be the line Trouble switching color, point M right over here, and I'm gonna construct a right triangle between point M and our center of rotation where the hypotenuse Gonna take each point, let's say point, using But how do we do that? Well, like we've done in previous videos what I'm gonna do is I'm So, let's just, instead of thinking of this in terms of rotating 270 degrees in the positive direction, in the counter-clockwise direction, let's think about, let's think about this, rotating this 90 degrees And 90 degree rotations are a little bit easier to think about. You see that that is equivalent, that is equivalent to a 90 degrees, to a 90 degrees clockwise rotation, or a negative 90 degree rotation. We're going in aĬounter-clockwise direction. If you imagine a point right over here this would be 90 degrees, 180, and then that is 270 degrees.
![rotation rules geometry clockwise rotation rules geometry clockwise](https://useruploads.socratic.org/adpoGJ2HRpqQlgK7eZWM_circle1.jpg)
Negative 270 degree rotation, but if we're talking aboutĪ 270 degree rotation. In the previous video when we were rotating around the origin, if you rotate something by, last time we talked about a So, to help us think about that I've copied and pasted this on my scratch pad and we can draw through it and the first thing that we might wanna think about is if you rotate, I've talked about this So, this would be 270 degrees in the counter-clockwise direction. The direction of rotation by a positive angle is counter-clockwise. We have this little interactive graph tool where we can draw points or if we wanna put them in the trash we can put them there.
![rotation rules geometry clockwise rotation rules geometry clockwise](https://i.stack.imgur.com/T0zwl.png)
Triangle SAM, S-A-M, and this is one over here, S-A-M, is rotated 270 degrees, about the point four comma negative two. But it's easy to calibrate it if you want to specify another point, around which you want to rotate - just make that point the new origin! Figure out the other points' coordinates with respect to your new origin, do the transformations, and then translate everything back to coordinates with respect to the old origin. Now, since (a, b) are coordinates with respect to the origin, this only works if we rotate around that point. Rotating (a, b) 360° would result in the same (a, b), of course. Because the axes of the Cartesian plane are themselves at right angles, the coordinates of the image points are easily predictable: with a bit of experimentation, you could easily 'prove' to yourself that rotating (a, b) 90° would result in (-b, a) rotation of 180° gives us (-a, -b) and one of 270° would bring us to (b, -a). So from 0 degrees you take (x, y) and make them negative (-x, -y) and then you've made a 180 degree rotation.Here's my idea about doing this in a bit more 'mathematical' way: every rotation I've seen until now (in the '.about arbitrary point' exercise) has been of a multiple of 90°. When you rotate by 180 degrees, you take your original x and y, and make them negative. If you have a point on (2, 1) and rotate it by 180 degrees, it will end up at (-2, -1) We do the same thing, except X becomes a negative instead of Y. If you understand everything so far, then rotating by -90 degrees should be no issue for you. Our point is as (-2, -1) so when we rotate it 90 degrees, it will be at (1, -2)Īnother 90 degrees will bring us back where we started. What about 90 degrees again? Same thing! But remember that a negative and a negative gives a positive so when we swap X and Y, and make Y negative, Y actually becomes positive. Our point is at (-1, 2) so when we rotate it 90 degrees, it will be at (-2, -1) What if we rotate another 90 degrees? Same thing. So from 0 degrees you take (x, y), swap them, and make y negative (-y, x) and then you have made a 90 degree rotation. When you rotate by 90 degrees, you take your original X and Y, swap them, and make Y negative. If you have a point on (2, 1) and rotate it by 90 degrees, it will end up at (-1, 2) In case the algebraic method can help you: