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The function in question appears in lemma $6.3$ of Milnors characteristic classes. Let $\langle \cdot, \cdot\rangle$ denote the dot product on $\mathbb{R}^m$

Let $u,v \in \mathbb{R}^m$ be unit vectors with $u \neq -v$. Let $T(u,v)$ denote the unique rotation of $\mathbb{R}^m$ which carries vector $u$ to vector $v$ and leaves everything orthogonal to $u$ and $v$ fixed. Alternatively, $T(u,v)$ can be defined by:

$$T(u,v)(x)=x - \frac{\langle(u+v),x\rangle}{1+\langle u,v\rangle}(u+v)+2\langle u,x\rangle v$$

Can somebody help me understand why this formula above works? I'm having troubles understanding why this formula gives the desired rotation. Thank you.

  • See my edits for proper MathJax usage. I changed $T(u,v)(x)=x - \frac{<(u+v),x>}{1+< u,v>}(u+v)+2<u,x> v$ to $$ T(u,v)(x)=x - \frac{\langle(u+v),x\rangle}{1+\langle u,v\rangle}(u+v)+2\langle u,x\rangle v $$ and and $<-,->$ to $\langle\cdot,\cdot\rangle,$ and did some other emendations. You can google "latex symbols" in such cases. – Michael Hardy Mar 25 '21 at 23:52
  • If you're doing this kind of math, you ought to refer to a LaTeX manual when doing things like this. – Michael Hardy Mar 25 '21 at 23:55
  • @MichaelHardy I found the book online. The missing hypothesis is that $u,v$ are unit vectors. I pasted the relevant paragraph into an answer – Will Jagy Mar 26 '21 at 00:11
  • @WillJagy : Is your comment supposed to be a reply to one of mine? And if not, why did you ping me? – Michael Hardy Mar 27 '21 at 05:17
  • @MichaelHardy just that the vectors were actually required to be unit vectors; you were already involved with the content of the question as far as edits. Apparently the OP edited in "unit vectors" – Will Jagy Mar 27 '21 at 13:19

3 Answers3

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Note $\|u+v\|^2=2(1+\langle u,v\rangle)$. If we write $\displaystyle w=\frac{u+v}{\|u+v\|}$, the given formula is

$$ R = \mathrm{Id}-2ww^T+2vu^T. $$

Where does this formula come from?

One clue is that $\mathrm{Id}-2ww^T$ is the Householder matrix for reflection across $w$'s orthogonal complement. Indeed, a good geometry exercise is that the product of two hyperplane reflections is a plane rotation by $2\theta$, in the plane spanned by the hyperplanes' two normals, where $\theta$ is the angle between the hyperplanes. One can argue it suffices to do this in 2D, and then it becomes a synthetic geometry exercise.

Note that $w$ is halfway between $u$ and $v$, so the angle $\angle uw$ is half the angle $\angle uv$. Therefore we may take the product of the two Householder reflections associated with $u,w$'s complements:

$$ (\mathrm{Id}-2ww^T)(\mathrm{Id}-2uu^T). $$

If you FOIL this out, then simplify with $w=u+v$, you get $R$ (exercise).

anon
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Page 77 in Milnor and Stasheff. They are required to be unit vectors.

enter image description here

Will Jagy
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Hints: Using $A$ for the given linear map and assuming $u$ and $v$ are unit vectors.

  1. If $x\perp u,v$ then $A(x)=x$.
  2. $A(u)=v$.
  3. $A$ preserves norm and inner product in the span of $u,v$.
  4. Deduce that $A$ is an orthogonal transformation.
Berci
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    they really are unit vectors. the OP neglected to state this. I pasted the correct page into an answer. – Will Jagy Mar 26 '21 at 00:28
  • Ah. Oops. I did indeed notice this when I typed it up, but I thought Milnor was just generalizing the word rotation or something >.< –  Mar 26 '21 at 00:33