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## Basic geometry and measurement

# Rotating points

Positive rotation angles mean we turn counterclockwise. Negative angles are clockwise. We can think of a 60 degree turn as 1/3 of a 180 degree turn. A 90 degree turn is 1/4 of the way around a full circle. The angle goes from the center to first point, then from the center to the image of the point.

## Want to join the conversation?

- So just to confirm what Sal briefly explained in the video, a
**positive number**of degrees means you rotate*counterclockwise*, while a**negative number**of degrees means you rotate*clockwise*, correct?(78 votes)- Seems a bit un-intuitive but since the 1st quadrant starts from right and goes to left, adding
**positive number**of degree means you go towards left i.e. you rotate*counter-clockwise*and vice versa.(43 votes)

- Are there formulas to do 90,180,270, rotations around a point that is not the origin?(12 votes)
- Four years late, but here it is.

Yes, there are. I was learning about this to help with a diagnostic. All you have to do is make the point you´re rotating around the new origin. You might have to tilt the whole coordinate plane, or even chop off parts of it and make new areas.(6 votes)

- Why do you have to rotate counterclockwise if the angle is

positive, and clockwise if the angle is negative?(10 votes)- It goes back to the unit circle where 0 degrees is along the positive x axis. All our angles greater than 0 and less than 90 (positive y axis) are in the first quadrant (positive x, positive y), so this is that counterclockwise rotation.(14 votes)

- Is rotating basically just eyeballing the correct angle?(12 votes)
- For these exercises, it is my impression that you are best served by eyeballing.

If you want to get more precise, you would use an instrument that measures angles (the most common example is a protractor) and verify that your point-to-point mappings satisfy the rotation angle requirement. You would also want to make sure that distances from the point of rotation are the same. Most protractors also have a limited ruler along the edge.(8 votes)

- Why is that when he was talking about going to the positive direction, he went left, and when he was talking about the negative direction he when right. Shouldn't it be the opposite way around?(6 votes)
- A positive rotation is in the counterclockwise direction. With unit circle theory, the positive x axis is 0 degrees, so rotating into the first quadrant gives positive values for sin and cos which make best sense for angles between 0 and 90.(9 votes)

- Why would POSITIVE 60 mean that your going counterclockwise?(5 votes)
- If you think about the positioning of the quadrants, it goes counterclockwise. That is probably why it goes counterclockwise with a positive. I DO NOT KNOW IF THIS IS RIGHT, so please ignore this post(5 votes)

- @khan what about the negative rotations?(4 votes)
- A positive rotation tells you to go counter-clockwise while a negative rotation tells you to go clockwise(6 votes)

- Are we just supposed to guess/assume the degrees in rotation? At the beginning of the video Sal said “This looks like 60 degrees” so are we just guessing?(5 votes)
- It is not guessing, it is more estimating which are different processes.(3 votes)

- Sal: Makes an explainer video to explain rotation

Sal: Doesn't explain rotation.

Positive rotation is going in the negative/counter-clockwise direction. You forgot to actually explain that, haha.(6 votes) - How do you find out the angle if the two points are close together?(4 votes)

## Video transcript

- [Instructor] We're told
that point P was rotated about the origin (0,0) by 60 degrees. Which point is the image of P? Pause this video and see
if you can figure that out. All right, now let's think about it. This is point P. It's being rotated around the origin (0,0) by 60 degrees. So if originally point
P is right over here and we're rotating by positive 60 degrees, so that means we go counter
clockwise by 60 degrees. So this looks like about
60 degrees right over here. One way to think about 60 degrees, is that that's 1/3 of 180 degrees. So does this look like 1/3 of 180 degrees? Remember, 180 degrees would
be almost a full line. So that indeed does look
like 1/3 of 180 degrees, 60 degrees, it gets us to point C. And it looks like it's the same distance from the origin. We have just rotated by 60 degrees. Point D looks like it's more
than 60 degree rotation, so I won't go with that one. All right, let's do one more of these. So we're told point P was
rotated by negative 90 degrees. The center of rotation is indicated. Which point is the image of P? So once again, pause this video and try to think about it. All right, so we have
our center of rotation, this is our point P, and we're rotating by negative 90 degrees. So this means we are going clockwise. So we're going in that direction. And 90 degrees is easy to spot. It's a right angle. And so it would look like
that and it looks like it is getting us right to point A. So this is a negative 90 degree
rotation right over here. Gets us to point A.