19. Hyperlining

Let $P=(p_1,p_2,\ldots,p_n)$ and $Q=(q_1,q_2,\ldots,q_n)$ be two points in $\mathbb{R}^n$.

1. Derive and prove a general formula for the Euclidean distance between $P$ and $Q$ in $\mathbb{R}^n$.

2. Let $A\in\mathbb{R}^n$ be a point and let $L=\{A+t\mathbf{v}:t\in\mathbb{R}\}$ be a line, where $\mathbf{v}\neq\mathbf{0}$. Using the distance formula from part (1), derive a general formula for the shortest distance between an arbitrary point $P\in\mathbb{R}^n$ and the line $L$.

Justify all steps in your derivation.