WEBVTT 00:00:10.056 --> 00:00:15.396 Spatial skills that students should be developing is their ability to mentally rotate objects, 00:00:15.876 --> 00:00:20.856 and so we spent a lot of time talking about how to rotate an object, 00:00:20.856 --> 00:00:24.836 how to think about rotating an object in our spatial skills training class. 00:00:25.236 --> 00:00:29.316 And what we usually start out talking about is that you have an object here 00:00:30.156 --> 00:00:34.126 and you have to think-and you've got three coordinate axes listed here. 00:00:34.556 --> 00:00:40.056 And you want to think about rotating that object about each one of these axes, 00:00:40.056 --> 00:00:43.586 and as you rotate this object about the various axes, 00:00:43.586 --> 00:00:45.486 you're going to get different views of that object. 00:00:45.956 --> 00:00:52.376 So if you start with the object here and you rotate it about this axis right here-if you think 00:00:52.376 --> 00:00:58.106 about the way an object rotates, it rotates like this in space. 00:00:58.186 --> 00:01:04.965 And so for this particular object, if it rotates about the-what I'm going to call the X-axis, 00:01:05.325 --> 00:01:09.306 it kind of falls forward and it ends up looking like this. 00:01:10.546 --> 00:01:17.116 If however, you start with the same-the original object and instead of rotating 00:01:17.116 --> 00:01:23.236 about this X-axis, you rotate it about the Y-axis-the vertical axis here-in this way, 00:01:23.516 --> 00:01:26.866 you'll end up with a view of it looking like this instead. 00:01:28.066 --> 00:01:32.126 And if you go back to the original object and you think about rotating 00:01:32.176 --> 00:01:37.286 about this axis here-the Z-axis, you think about-it kind of falls backwards that way, 00:01:37.506 --> 00:01:40.956 and this is, then, what you end up with as your view of that object. 00:01:40.956 --> 00:01:46.646 So you can see that we started with the same object, and depending upon which axis we chose 00:01:46.646 --> 00:01:50.256 to rotate about, we got three entirely different views of that object. 00:01:50.646 --> 00:01:55.936 And that's what you want the students to understand is that they need to be able to think 00:01:55.936 --> 00:01:59.066 about what this would look like if I rotate it about one 00:01:59.066 --> 00:02:01.346 of these axes or about both of these axes. 00:02:02.316 --> 00:02:09.306 Now, in order to make our lives easier-in math and in science and in engineering 00:02:09.306 --> 00:02:13.076 and in those kinds of fields, we talk about positive and negative rotations. 00:02:13.566 --> 00:02:21.256 So a positive rotation is if you think about it, if you were looking down the particular axis 00:02:21.546 --> 00:02:25.546 and it was counterclockwise, that would be a positive rotation. 00:02:25.956 --> 00:02:29.016 If it was clockwise, it would be a negative rotation. 00:02:29.466 --> 00:02:32.356 So here we have an object that starts out looking like this, 00:02:32.356 --> 00:02:37.196 we're going to rotate-if we rotate it in a positive direction about this particular axis, 00:02:37.516 --> 00:02:40.636 you would end up with an object that looks like this one right here. 00:02:41.636 --> 00:02:44.476 Now if you think about looking down this axis, 00:02:45.206 --> 00:02:50.036 that is a counterclockwise rotation about that axis. 00:02:51.236 --> 00:02:55.386 If you think about a negative rotation, which is a clockwise rotation 00:02:55.856 --> 00:02:59.426 and you take this original object, it would end up looking like this. 00:02:59.426 --> 00:03:04.306 And now again, you're looking down that axis and it's going clockwise 00:03:04.306 --> 00:03:07.026 in this case, and that is a negative rotation. 00:03:08.436 --> 00:03:12.286 One of the things that we use in engineering and in science 00:03:12.286 --> 00:03:18.536 and math a lot is the right hand rule to describe rotations, rather than the clockwise 00:03:18.536 --> 00:03:21.596 and counterclockwise rotation I just talked about. 00:03:21.726 --> 00:03:25.736 A positive rotation-for a positive rotation, 00:03:25.806 --> 00:03:32.066 you imagine that you put your thumb down the axis of interest. 00:03:32.606 --> 00:03:37.816 And your fingers will curl in the direction of that positive turn. 00:03:37.816 --> 00:03:39.346 This is called the right hand rule. 00:03:39.346 --> 00:03:42.316 So you put the thumb of your right hand down the axis, 00:03:42.376 --> 00:03:46.126 your fingers curl in the direction of the rotation-that's the right hand rule. 00:03:46.476 --> 00:03:52.866 Notice that if you look at this-if you think about the axis coming straight out at you 00:03:52.866 --> 00:03:57.596 and you curl your fingers-that is, indeed, a counterclockwise rotation about the axis. 00:03:57.896 --> 00:04:02.256 But being able to use it with your right hand is sometimes easier for students 00:04:02.296 --> 00:04:04.306 to visualize what that looks like. 00:04:04.686 --> 00:04:06.046 And, of course, if it's negative, 00:04:06.046 --> 00:04:09.616 your hands are going to curl in the other-your fingers will curl in the other direction. 00:04:10.186 --> 00:04:13.456 So I have a problem here that we're going to work on. 00:04:13.456 --> 00:04:15.726 We're going to do a couple rotation problems. 00:04:17.016 --> 00:04:20.076 Here we have an axis shown like this. 00:04:20.076 --> 00:04:23.006 We have our object built like this. 00:04:23.366 --> 00:04:29.706 And now we want to do a negative, 90 degree rotation about the axis that's shown. 00:04:30.086 --> 00:04:35.316 So if we want to do a negative, 90 degree rotation-again, 00:04:35.316 --> 00:04:39.866 we're going to point the thumb of our right hand down the negative part of that axis, 00:04:40.196 --> 00:04:44.526 curl our fingers in that direction, so this object is going to fall over this way 00:04:44.706 --> 00:04:48.816 when we do a negative, 90 degree rotation about that particular axis. 00:04:49.156 --> 00:04:53.606 So after I've rotated that in space, what does this object look like? 00:04:54.266 --> 00:04:56.756 Well, it's going to look like this. 00:04:57.556 --> 00:05:18.836 We sketch it...so that's what the object looks like after it has been rotated 00:05:18.836 --> 00:05:21.846 about this axis by a minus 90 degrees. 00:05:22.936 --> 00:05:29.106 Here we have an object that we want to rotate a positive 180 degrees. 00:05:29.456 --> 00:05:32.256 Again, I have the object built out of blocks here. 00:05:32.256 --> 00:05:35.426 You can see I've got it lined up kind of along the axis. 00:05:35.816 --> 00:05:40.506 I'm going to point my thumb down the positive direction, now, of that axis, 00:05:40.696 --> 00:05:43.056 and I'm going to curl my fingers in this direction, 00:05:43.056 --> 00:05:47.996 and it's going to rotate 90 degrees and then 180 degrees. 00:05:48.266 --> 00:05:52.046 And so now I would have students sketch what this looks 00:05:52.046 --> 00:05:56.006 like after it's been rotated in space 180 degrees. 00:05:56.806 --> 00:06:10.296 Start with the back surface here...and so that would be the answer to that particular problem. 00:06:10.296 --> 00:06:14.526 Then we have a number of problems that we work where students have to rotate first 00:06:14.526 --> 00:06:18.206 about one axis and then the other, but these are the simplest ones, 00:06:18.206 --> 00:06:19.756 and that's what we've started with.