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## Homework Statement

Let [itex] c(s) = \left( \begin{array}{ccc}

\cos(s) & -\sin(s) & 0 \\

\sin(s) & \cos(s) & 0 \\

0 & 0 & 1 \end{array} \right) [/itex] be a curve in SO(3). Find the tangent vector to this curve at [itex] I_3 [/itex].

## Homework Equations

Presumably, the definition of a tangent vector as a differential operator would be useful here:

If X is the tangent vector to [itex] c(s) [/itex] at [itex] I_3 [/itex] (i.e. [itex] s = 0 [/itex]), then for functions [itex] f: SO(3) \rightarrow \mathbb{R} [/itex],

[itex] X\left[ f \right] = \frac{df\left(c\left(s\right)\right)}{ds}|_{s=0} [/itex]

## The Attempt at a Solution

The fact that this problem is found in the section of the textbook (Nakahara,

__Geometry, Topology and Physics__) dealing with Lie groups and Lie algebras, together with the fact that the problem asks for the tangent vector to c(s) at the identity leads me to think that we are supposed to use the Lie algebra in some way. However, I can't see how one would do that, so I've tried proceeding naively from the definition:

[itex] X\left[ f \right] = \frac{df\left(c\left(s\right)\right)}{ds}|_{s=0} [/itex]

[itex] = \frac{df}{dc} \frac{dc}{ds}|_{s=0} [/itex]

[itex] = \frac{df}{dc} \left( \begin{array}{ccc}

0 & -1 & 0 \\

1 & 0 & 0 \\

0 & 0 & 0 \end{array} \right) [/itex]

I have no idea how to proceed from here.

Thanks for any and all help!

EDIT: I quickly realized how trivial this is.

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