What exactly does a rotation preserve?

I understand a rotation should preserve length and angle and hence the dot product. Since anything that preserves the dot product is a linear transformation, then a rotation can be represented by a matrix. Let's only consider real matrices.

However, there is something else that a rotation preserves and I am here to ask what it is. A linear transformation that preserves length must be orthogonal, however, an orthogonal matrix is a rotation if its determinant is $1$. That means a rotation is a special case of orthogonal matrix.

So what makes a rotation special from orthogonal matrices? What else does a rotation preserve in addition to length and angle? Anyone can provide a prove that a matrix is a rotation iff it is an orthogonal matrix with determinant $1$?

Thank you!

• A rotation preserves the origin as a fixed point and it preserves orientation. Jun 19, 2016 at 19:57
• Yes you mention determinant 1, that is special. If it was allowed to have -1 it would end up "mirroring" something. Jun 19, 2016 at 19:58

A rotation preserve also the orientation of a figure. An orthogonal matrix has determinant $\pm 1$ and preserves lengths and angles, if the determinant is $=1$ it preserve also the orientation.