Consider the normed space $(\mathbb R^2, |\cdot|).$ Given a point $p\in\mathbb R^2$ and a subset $S\subset\mathbb R^2$, define the distance from $p$ to $S$ by $$d(p,S)=\inf\{|p-q|\colon q\in S\}.$$ Suppose $S$ is a line given by the equation $a\cdot x+b\cdot y+c=0$ (where $a,b,c\in\mathbb R$ are constant). How can I show that $$d(p,S)=\frac{|a\cdot x+b\cdot y+c|}{\sqrt{a^2+b^2}},$$ by using the definition of $d(p,S)$ (without using a linear algebra argument)?
How to determine the distance from a point to a line
1
$\begingroup$
geometry
-
0The formula you wrote down does not evaluate to a real number, but is a function of $x$ and $y$. – 2012-06-03
-
0You've made a mistake in your formula for $d(p,S)$. There should be $\sqrt{a^2+b^2}$ in the denominator – 2012-06-03
-
0The line formula is for $\mathbb{R}^2$. So, you should replace the $n$ by $2$. – 2012-06-03