# symmetry-rotation-vector-invariants.mp4

Chapter section: Invariants and Metrics

Caption (verbatim):

Metric structure

Image description (verbatim):

Rotation preserves vector lengths and angles

## Before the animation

Chapter lines 341–341:

What is the invariant of rotation in this representation? It is just the length of vectors and the angles between them.

## After the animation

Chapter lines 349–349:

These relationships are expressed by a single invariant, the dot product.

Chapter lines 351–355:

```math
\mathbf u\cdot\mathbf v
=
u_xv_x+u_yv_y.
```