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Here of course you're supposed to look at them from above (e.g. $^1$One example of this is hurricanes, which are said to rotate (anti)clockwise in the (northern) southern hemisphere.
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Given a directed axis, put your right hand with your thumb pointing along the axis in the given direction, and then your fingers will point to the "positive" direction of rotation. It must also be noted that the "positive" direction of rotation is usually defined to be anticlockwise. Put your left hand on the page with your thumb toward you, and your fingers will be clockwise. This is equivalent to setting the plane normal "up", towards you.
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Put the page on the wall so you can read it, next to a clock whose face you can see its hands will move clockwise. when doing 2D analytical geometry), then there is a canonical way to understand the term. In your terms,Īdditionally, if the question is purely about a planar problem (i.e. It coincides with the direction your fingers point to when you put your left hand in front of you with your thumb pointing towards you. This refers to the direction of rotation of the hands of a (normal, non-transparent) clock as you observe them. That said, a statement like "clockwise as seen from the top" is not ambiguous. (In more technical language, "clockwise" defines an order for two basis vectors for the plane, but does not specify a sign for the normal.) If you take a transparent clock mounted on a glass window, it will rotate clockwise or anticlockwise depending on what side of the window you're standing. The reason for this is that "clockwise" defines a direction of rotation within a plane, but does not specify which side the plane is observed from. The state of the x axis remains unchanged - we've started with a state of no rotation so the x axis will retain its original state - the unit vector $\left[\begin$, and now you'll notice that the transpose of the counterclockwise representation gives you the clockwise representation! Of course, rotating clockwise and rotating counterclockwise by $\phi$ radians are inverse operations.Your confusion arises because the term (anti)clockwise, when used by itself, is ambiguous, and should always be used with a statement like "as seen from the top" (unless that is absolutely obvious$^1$).If we start with the "no rotation has taken place" stateĪnd want to rotate around - let's say - the x axis we will do Whenever we want to rotate around an axis, we are basically saying "The axis we are rotating around is the anchor and will NOT change. Usually I am a fan of explaining such things in 2D however in 3D it is much easier to see what is happening. Here we are using the canonical base for each space that is we are using the unit vectors to represent each of the 2 or 3 axes. This means that no rotation has taken place around any of the axes.Īs we know $\cos(0) = 1$ and $\sin(0) = 0$.Įach column of a rotation matrix represents one of the axes of the space it is applied in so if we have 2D space the default rotation matrix (that is - no rotation has happened) isĮach column in a rotation matrix represents the state of the respective axis so we have here the following:įirst column represents the x axis and the second one - the y axis. This matrix can represent a rotation around all three axes in 3D Euclidean space with. Here is a "small" addition to the answer by you have the following rotation matrix:Īt first one might think this is just another identity matrix.