# Continuously rotating a unit vector to $e_1$

The following question is from Brian C. Hall's Lie Groups, Lie Algebras, and Representations.

Show that $$\mathrm{SO}(n)$$ is connected, using the following outline.

For the case $$n =1$$, there is nothing to show, since a $$1 ×1$$ matrix with determinant one must be $$[1]$$. Assume, then, that $$n\ge 2$$. Let $$e_1$$ denote the unit vector with entries $$1, 0,\ldots,0$$ in $$\Bbb R^n$$. For every unit vector $$v \in \Bbb R^n$$, show that there exists a continuous path $$R(t) \in \mathrm{SO}(n)$$ with $$R(0)=I$$ and $$R(1)v =e_1$$. (Thus, any unit vector can be “continuously rotated” to $$e_1$$.)

I'm aware of alternate proofs, but I'm looking for $$R (t)$$ mentioned above.

I know that $$\mathrm{SO}(n)$$ acts transitively on $$S^{n-1}$$. So there exists a matrix $$A\in \mathrm{SO}(n)$$ such that $$Av=e_1$$. Define $$R:[0,1]\to \mathrm{SO}(n)$$ by $$R (t)=(1-t)I+tA$$. There are two things I need to verify; $$R (t)^{\rm T}R (t)=I$$ and $$\det R (t)=1.$$

But $$R (t)^{\rm T}R (t)=((1-t)I+tA^{\rm T})((1-t)I+tA )\\=(1-t)^2 I+(1-t)tA+(1-t)tA^{\rm T}+t^2I.$$

Any hints on how to find the $$R(t)$$? Thank you.

Your $$R(t)$$ is not necessarily orthogonal.
Suppose $$e_1$$ and $$v$$ are elements of a two-dimensional subspace $$U$$ with orthonormal basis $$(e_1,w)$$. Then $$v=\cos\theta\, e_1+\sin\theta\, w$$ for some theta. Let $$R(t)$$ map $$e_1$$ to $$\cos t\theta\, e_1+\sin t\theta\,w$$ and $$w$$ to $$-\sin t\theta\,e_1+\cos t\theta\, w$$ and also restrict to the identity on $$U^\perp$$, the orthogonal complement of $$U$$.
• $v=\cos\theta\, e_1+\sin\theta\, w$ because $v$ is a unit vector, am I right ? – Thomas Shelby May 19 at 16:12