1. Shortest distance

how do you prove that the shortest distance from a point to a line ( $L$) in $R^3$ (or any $R^n$) is achieved along the perpendicular line connecting A to line( $L$).

thanks

2. On the assumption that a unique perpendicular $RP$ from point $R$ of line $L$ to point $P$ always exists, you may consider $\bigtriangleup PQR$, where $
Q$
is any other point on line $L$.

3. this is an assumption
Originally Posted by Scott H
that a unique perpendicular $RP$ from point $R$ of line $L$ to point $P$ always exists
how do we prove this though?

i understand that i must use the dot product and it should equal zero, where the shortest distance is when the point( $A$) is perpendicular to the line ( $L$)

4. proved for two-dimensional space and then saw R3..OOPS!