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Let's say I have a vector $\overrightarrow{a}$ (as shown by the arrow in the picture) with a certain angle $\theta$ relative to $x$-axis, which denotes a tangent line of a circle $C$ with radius $r$. The goal is to rotate $\overrightarrow{a}$ relative to the midpoint of the circle $C$ so that $\overrightarrow{a}$ directly points to the point $P(x,y)$ (denoted by the red square). By what angle $\phi$ should my arrow now be rotated?

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gilianzz
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    Do you know how to find the tangent on a circle from an external point? If yes, do so, find out the angle with $x$ axis, and you will have the elements for your answer... If no, look it up! – Martigan Jun 15 '18 at 08:10
  • The usual interpretation of a vector is that it has only a length and direction, not a fixed starting position. Rotating a vector changes its direction, nothing else, and it does not matter what point you rotate around. It seems you may be interested in an ordered pair of points (the “beginning” and “end” of the arrow) rather than the usual idea of a “vector” in a plane. – David K Jun 15 '18 at 08:13
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    For finding the angle to the tangential point draw a line from the center of the circle to the point. Draw a line from the tangent point of the circle to the point. Notice that you have a 90 degree angle in that triangle and you have two distances. Start with this and see if you can find your way towards the solution. – Stefan Jun 15 '18 at 08:26
  • The “vector” isn’t uniquely determined by your description. There are two possibilities. – amd Jun 17 '18 at 08:46

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