 # Calculate angle between two vectors, given their rotation w.r.t. a third vector. I have three vecto Kolten Conrad 2022-07-08 Answered
Calculate angle between two vectors, given their rotation w.r.t. a third vector.
I have three vectors, $\stackrel{\to }{a},\stackrel{\to }{b},$ and $\stackrel{\to }{c}$ in n-dimensional space. I know the coordinates of all three vectors and their dot products. Both $\stackrel{\to }{a}$ and $\stackrel{\to }{b}$ are rotated away from $\stackrel{\to }{c}$ by an angle $\alpha$, in their own respective directions, obtaining ${\stackrel{\to }{a}}^{\prime }$ and ${\stackrel{\to }{b}}^{\prime }$ . What is the angle between ${\stackrel{\to }{a}}^{\prime }$ and ${\stackrel{\to }{b}}^{\prime }$ ?
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Step 1
Assuming that a,b,c have norm 1, we have
$\begin{array}{rl}{a}^{\prime }& =2\left(a\cdot c\right)a-c\\ {b}^{\prime }& =2\left(b\cdot c\right)b-c\end{array}$
Indeed, this implies ${a}^{\prime }\cdot a=a\cdot c$ and ${a}^{\prime }\cdot c=2\left(a\cdot c{\right)}^{2}-1$ which is the cosine of the double angle. Alternatively, it is obvious geometrically that $\frac{{a}^{\prime }+c}{2}$ is the orthogonal projection of c on a, that is to say $\left(a\cdot c\right)a$
Hence
${a}^{\prime }\cdot {b}^{\prime }=4\left(a\cdot c\right)\left(b\cdot c\right)\left(a\cdot b\right)-2\left(a\cdot c{\right)}^{2}-2\left(b\cdot c{\right)}^{2}+1$

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