1. Shortest distance

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

thanks

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

3. this is an assumption
Originally Posted by Scott H
that a unique perpendicular $\displaystyle RP$ from point $\displaystyle R$ of line $\displaystyle L$ to point $\displaystyle P$ 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($\displaystyle A$) is perpendicular to the line ($\displaystyle L$)

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