Orthogonal Matrix

A square matrix with real numbers or elements is orthogonal if its transpose is equal to its inverse:

$$A^T=A^{-1}$$

or

$$A{A}^T=A^{T}A=I$$

Geometric Meaning

For any two vectors, $u,v\in {\mathbb{R}}^n$, their dot product after applying an orthogonal matrix $A$ is:

$$(Au)\cdot (Av)=u^T{A}^{T}Av=u^{T}v=u\cdot v$$

So, the dot product is preserved; is is the same whether you apply the matrix or not.