# How to calculate the rotation of a vector?

So, let's say I have vector $\vec{ab}$ and vector $\vec{ac}$. How do I calculate the amount of rotation from $b$ to $c$?

Note, this is in a 3D space, of course...

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Use the formula:

$\cos \theta = \frac{\vec {ab}\cdot \vec{ac}}{|\vec{ab}||\vec{ac}|}$

where $\theta$ is the angle between $\vec{ab}$ and $\vec{ac}$.

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Alright, thanks! – Steven Fontaine Jan 1 '13 at 0:06
Wait.. I just thought about something.. Would it just be $\cos \theta = \frac{\vec {b}\cdot \vec{c}}{|\vec{b}||\vec{c}|}$ or do I have to do something with $a$? – Steven Fontaine Jan 1 '13 at 0:25
@Steven, what you just commented on is just an issue of notation. Basically, you should focus on the 2 vectors and the angle between them. You can call the vectors $\vec{b}$ and $\vec{c}$ or (if you want to describe the vectors in terms of points a, b & c) $\vec{ab}$ and $\vec{ac}$. Does that help? – Conan Wong Jan 1 '13 at 0:31
Okay, yes. Thanks again. – Steven Fontaine Jan 1 '13 at 0:46
If you don't acknowledge $a$, the point being rotated around, how can you figure the angle? I haven't actually tried it yet, but I don't understand how this could possibly work, since there's an infinite amount of possible points that the point rotated could have been rotated around. (Those points being contained in a line from the center of the circle being rotated around and travels perpendicular to the circle) So, how can this work? – Steven Fontaine Jan 1 '13 at 5:17

Let $\,\theta\,$ be the angle between the given vectors: $\,\vec{ab}\;\text{and}\; \vec{ac}\,.$

Recall that $$\cos \theta\; = \;\left(\frac{(\vec{ab})\cdot (\vec{ac})}{|\vec{ab}||\vec{ac}|}\right)\;.$$

Solving for $\,\theta\,$ gives us: $$\theta \;= \;\cos^{-1}\left(\frac{(\vec{ab})\cdot (\vec{ac})}{|\vec{ab}||\vec{ac}|}\right)$$

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Oh, so I don't use $\vec{ac}$, I use $\vec{bc}$? – Steven Fontaine Jan 1 '13 at 5:21
That's correct, Steven - typo (mind wandering: just answered a question on transitivity: $(a, b) \land (b, c) \implies ...$). The angle of rotation to get from b to c is the angle between $\vec {ab}$ and $\vec {ac}$ – amWhy Jan 1 '13 at 19:06