# 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. – almagest Jun 19 '16 at 19:57
• Yes you mention determinant 1, that is special. If it was allowed to have -1 it would end up "mirroring" something. – mathreadler Jun 19 '16 at 19:58

## 1 Answer

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.

• Thank you for your answer. Is there any formal and general definition of "orientation"? – Ralph B. Jun 19 '16 at 20:01
• Orientability is a property of a figure that, roughly speaking, gives the possibility of distinguish a clockwise or counterclockwise in a loop in the figure, so that we can speak of a clockwise or counterclockwise orientation . A rogorous definition can be done for manifolds of any dimension. You can see here en.wikipedia.org/wiki/Orientability – Emilio Novati Jun 19 '16 at 20:09