Consider the set $P\left(\mathbb{R}^{n}\right)$ of one-dimensional subspaces of $\mathbb{R}^{n}$, that is to say lines through the origin).
One way to define a distance on this set is to take, for lines $L_{1}, L_{2}$, the distance between $L_{1}$ and $L_{2}$ to be
$$
d\left(L_{1}, L_{2}\right)=\sqrt{1-\frac{|\langle v, w\rangle|^{2}}{\|v\|^{2}\|w\|^{2}}},
$$
where $v$ and $w$ are any non-zero vectors in $L_{1}$ and $L_{2}$ respectively.
Consider the set $P\left(\mathbb{R}^{n}\right)$ of one-dimensional subspaces of $\mathbb{R}^{n}$, that is to say lines through the origin).
One way to define a distance on this set is to take, for lines $L_{1}, L_{2}$, the distance between $L_{1}$ and $L_{2}$ to be
$$
d\left(L_{1}, L_{2}\right)=\sqrt{1-\frac{|\langle v, w\rangle|^{2}}{\|v\|^{2}\|w\|^{2}}},
$$
where $v$ and $w$ are any non-zero vectors in $L_{1}$ and $L_{2}$ respectively.