I'm working my way through Poole's Linear Algebra: A Modern Introduction, and I'm having some trouble fully wrapping my head around his solution to a problem regarding orthogonal matrices and their preservation of angles. The problem states:
If $Q$ is an $n\times n$ matrix such that the angels $\angle(Q\mathbf{x},Q\mathbf{y})$ and $\angle(\mathbf{x},\mathbf{y})$ are equal for all vectors $\mathbf{x}$ and $\mathbf{y}$ in $\mathbb{R}^n$, prove that $Q$ is an orthogonal matrix.
Poole's solution is as follows:
The given condition means that for all vectors $\mathbf{x}$ and $\mathbf{y}$,$$\frac{\mathbf{x}\cdot\mathbf{y}}{\Vert\mathbf{x}\Vert\,\Vert\mathbf{y}\Vert}=\frac{Q\mathbf{x}\cdot Q\mathbf{y}}{\Vert Q\mathbf{x}\Vert\,\Vert Q\mathbf{y}\Vert}$$Let $\mathbf{q}_i$ be the $i^\text{th}$ column of $Q$. Then $\mathbf{q}_i=Q\mathbf{e}_i$, where the $\mathbf{e}_i$ are the standard basis vectors. Since the $\mathbf{e}_i$ are orthonormal and the $\mathbf{q}_i$ are unit vectors, we have $$\mathbf{q}_i\cdot \mathbf{q}_j=\frac{\mathbf{q}_i\cdot \mathbf{q}_j}{\Vert\mathbf{q}_i\Vert\,\Vert\mathbf{q}_j\Vert}=\frac{\mathbf{e}_i\cdot \mathbf{e}_j}{\Vert\mathbf{e}_i\Vert\,\Vert\mathbf{e}_j\Vert}=\mathbf{e}_i\cdot \mathbf{e}_j=\begin{cases}0 & i\neq j \\ 1 & i=j\end{cases}$$ It follows that $Q$ is an orthogonal matrix.
Nearly all of this reasoning makes sense to me; given that the $\mathbf{q}_i$ are unit vectors, the fact that $Q$ follows logically from the requirement that $$\frac{\mathbf{e}_i\cdot\mathbf{e}_j}{\Vert\mathbf{e}_i\Vert\,\Vert\mathbf{e}_j\Vert}=\frac{Q\mathbf{e}_i\cdot Q\mathbf{e}_j}{\Vert Q\mathbf{e}_i\Vert\,\Vert Q\mathbf{e}_j\Vert}=\frac{\mathbf{q}_i\cdot\mathbf{q}_j}{\Vert\mathbf{q}_i\Vert\,\Vert\mathbf{q}_j\Vert}$$ and the definition of an orthogonal matrix. However, I'm not sure what actually requires that the $\mathbf{q}_i$ be unit vectors. The only other stipulation on $Q$ in the premise for the problem was that it be a square matrix, so I'm assuming that the condition that the $\mathbf{q}_i$ be unit vectors is somehow implicit in the relationship $\frac{\mathbf{x}\cdot\mathbf{y}}{\Vert\mathbf{x}\Vert\,\Vert\mathbf{y}\Vert}=\frac{Q\mathbf{x}\cdot Q\mathbf{y}}{\Vert Q\mathbf{x}\Vert\,\Vert Q\mathbf{y}\Vert}$ for all $\mathbf{x}$ and $\mathbf{y}$. Can anybody help me understand Poole's justification for the $\mathbf{q}_i$ being unit vectors?