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Performing rotations (old)

An older video where Sal uses the interactive widget to find the image of a line segment under a rotation. Created by Sal Khan.

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  • old spice man green style avatar for user Leonardo DiCaprio
    Why doesn't it specify which way to turn, like 90 degrees clockwise or counterclockwise. I never heard of a -90 degree rotation. Any clarifications?
    (19 votes)
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  • blobby green style avatar for user MyImmortalWolf
    How do you do this manually?
    (8 votes)
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    • mr pants teal style avatar for user Wrath Of Academy
      Apparently you're asking how would you do this not on KhanAcademy, but rotate a point using a pencil and paper? For a simple rotation like 90 degrees you can just plot the points relative to the x-axis instead of y-axis. For a more complicated rotation like 7 degrees, you'd need to use a protractor to measure the new angle. Or if you're manually doing it on a computer you'd need to use the sine/cosine of the new angle to get the x & y coordinates, relative to the old angle.
      (5 votes)
  • male robot johnny style avatar for user gonjamichael
    How do i do this with a pencil and a graph book
    (3 votes)
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    • piceratops ultimate style avatar for user Stefan Steinbeck
      Rotation on paper, like the on at , can be done on paper. You do need a thumbscrew compass (note: this does not mean the navigational compass, but the drawing tool) and protractor, f.e. a set square with integrated protractor.
      The process is a bit complicated. So I do a step by step explanation:
      1.) Place the needle point at the center of the rotation, in this case the origin of the coordinate plane.
      2.) Now enlarge the compass, so that the pencil lead shows to the point you would like to rotate.
      3.) Draw a circle. So this would be a circle with the middle at the center of rotation and the point somewhere on the circular line.
      4.) Draw a line from the original point to the center.
      5.) Now you need to find the rotated point. You need your rotation angle, in this case 90°. Measure the rotation angle from the line in 4.) with the protractor.
      6.) Draw a second line from the middle with the angle from 5.) to the first line.
      7.) Draw a point at the intersection of your line from 5.) and the circular line from 3.). This is your rotated pint.
      8.) Repeat steps 1-7 for for the other point or if you would have a more complex figure for all remaining points.
      8.) Connect the points and your are done.

      Sorry for this complicated explanation. As a none native speaker a still have my problems with the proper english words for certain mathematical procedures.
      (7 votes)
  • aqualine ultimate style avatar for user 2019cspencer
    How do I actually do this not just watch them do it?
    (0 votes)
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  • aqualine tree style avatar for user Najla Ali
    Where did the name math come from?
    (2 votes)
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    • leafers seedling style avatar for user winola One
      The word mathematics comes from the Greek μάθημα (máthēma), which, in the ancient Greek language, means "that which is learnt",[24] "what one gets to know", hence also "study" and "science", and in modern Greek just "lesson".——from Wikipedia.
      (4 votes)
  • blobby green style avatar for user monte.ball.2018
    Why doesn't it tell you which way to turn it clockwise or counterclockwise. Is that even a thing, a 90 degree turn?
    (2 votes)
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  • piceratops ultimate style avatar for user thebaliguns
    So is it true that we have to use formulas to perform rotations? A lot of the comments are suggesting the use of formulas
    (2 votes)
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    • piceratops ultimate style avatar for user Barrett Southworth
      It's true that you have to use it in the exercises or at least some of them. There is also the little rotate tool at times. If you know the numerical coordinates of something you can figure out what they will be after a certain rotation. It is also possible to perform rotations at least approximately using just a compass and a protractor in the physical world.

      With the line segment problem, you can view the rotation about a certain point as being the center of the circles of the rotation. Choose one of the endpoints to rotate first. Create a circle which has the segment endpoint on it and the point being rotated about as the center. The rotated point will lie on this circle. The angle between the lines made by the two points on the circle and the center of the circle is the angle of rotation. Repeat this process for the other endpoint. Joining the two points creates the rotated line segment.
      (3 votes)
  • leafers sapling style avatar for user Sasha Aronson
    I'm thinking that there should be a way to rotate an object (in this case, a line segment) by using the coordinates alone in the event that a compass and pencil can't be used. For example, the top coordinate of the line segment before the shape is rotated is (-3,6). After rotating it, it becomes (6,3). I remember something about there being a rule for determining where the coordinates will end up depending on the type of rotation from its original position to its new position, but this video has made me unsure. Any thoughts?
    (2 votes)
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  • blobby green style avatar for user honorgy
    Where can I find this website so I can do that?
    (2 votes)
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  • leaf red style avatar for user я̿€̿ρ̿Ł̿ɪ̿т̿Ǻ̿ƶ̿
    Wouldn't it be alot harder if you didn't have a computer to rotate it for you? How would you even do that?
    (1 vote)
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    • piceratops ultimate style avatar for user Barrett Southworth
      I'd use a compass and protractor. I'd draw two circles to start with, both centered at the origin, with the outer edge being one of the line segment endpoints. I'd use the protractor, putting the middle dot at the origin of a circle and 0 at one of the line segment endpoints then measure out the rotation angle and plot that point on the same circle. After rotating both points I'd connect them to create the new line segment.
      (3 votes)

Video transcript

We're asked to rotate the line segment negative 90 degrees around the origin. So this is our line segment. We want to rotate it around the origin. So let's use our Rotate tool. So let's define our point of rotation. So we're going to rotate around the origin so that's why I put it there. And I'm going to rotate it negative 90 degrees. So I'm going to rotate it in this direction. So let's see if I can pull this off. So actually, let me undo what I just did. Let me get my Rotate tool back here. So negative 90 degrees. So these arrows right over here, right now they're straddling the x-axis. If I rotate negative 90 degrees, these arrows, this double-headed, curvy arrow should exactly straddle the negative y-axis. So let's make sure I do that. So this is the negative y-axis. There we go. So I have rotated this line segment negative 90 degrees about the origin. And now we've got to answer some questions. When a line segment is rotated, the image created is a, well, it's, of course, a line segment. We know this is true because the rotated image has two end points. The rotated image is the same length as the original image. We haven't changed the length of this, we just rotated it around this dot.