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Given two vectors in $\mathbb{R}^n$, $v_0$ and $v_1$, which define a plane including the origin a rotation along that plane can be defined from $v_0$ to $v_1$. I know the formula for rotation within a plane, but I'm getting confused on how to apply that arbitrary planar rotation to the other points in $\mathbb{R}^n$. The plane itself can be oriented in any direction depending on $v_0$ and $v_1$.

Mårten W
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sheppa28
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1 Answers1

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We may assume that $v_0$ and $v_1$ are non-parallel unit vectors. Let $u\in R^n$. Then $$\langle u,v_0\rangle v_0+\langle u,v_1\rangle v_1 $$ is the projection of $u$ in the plane. In that plane we rotate $v_0$ to $v_1$ and $v_1$ to $w=2\langle v_0,v_1\rangle v_1-v_0$. (Convince yourself that $v_1=v_0+w$ and $\|w\|=1$; it's helpful to draw a picture.)

So $$u\mapsto u-\langle u,v_0\rangle v_0-\langle u,v_1\rangle v_1 +\langle u,v_0\rangle v_1+\langle u,v_1\rangle(2\langle v_0,v_1\rangle v_1-v_0) $$ $$ =u-\langle u,v_0\rangle(v_0-v_1)-\langle u,v_1\rangle(v_1-w\rangle, $$ that is, we rotate the component of $u$ in the plane and leave the rest of $u$ unchanged.

Michael

Michael Hoppe
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  • Thanks for your quick reply, and apologies for the delay in responding. Can you please explain a little bit more? Confused on two points. #1. How v1=v0+w when w=2<v0,v1>v1-v0. Wouldn't that imply <v0,v1>=1/2? (add v0 to w and then v1=2<v0,v1)v1). #2 In the final equation is it <u,v0>v1 or <u,v0>v0? Thanks! – sheppa28 Sep 25 '13 at 20:32
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    To #1: you have to find a vector $w$ s.t. the bisector of $w$ and $v_0$ is $v_1$; thus $v_0$ is rotated to $v_1$ and $v_1$ is rotated by the same angle to that vector $w$; just draw a picture. Ad #2: The formula is correct as we want $v_0\mapsto v_1$ and $v_1\mapsto w$. – Michael Hoppe Sep 27 '13 at 05:30