Question

What is the mathematical definition of the distance between two vectors in $R^n$?

Answer 1

If $\mathbf{u}=\left(u_{1}, u_{2}, \ldots, u_{n}\right)$ and $\mathbf{v}=\left(v_{1}, v_{2}, \ldots, v_{n}\right)$ are points in $R^{n}$, then we denote the distance between $\mathbf{u}$ and $\mathbf{v}$ by $d(\mathbf{u}, \mathbf{v})$ and define it to be $$ d(\mathbf{u}, \mathbf{v})=\|\mathbf{u}-\mathbf{v}\|=\sqrt{\left(u_{1}-v_{1}\right)^{2}+\left(u_{2}-v_{2}\right)^{2}+\cdots+\left(u_{n}-v_{n}\right)^{2}} $$