# Thread: Finding the shortest distance from a point to a line with parametric vector equation?

1. ## Finding the shortest distance from a point to a line with parametric vector equation?

"Find the shortest distance from the point P = P(-1,0,2) to the line given by the parametric vector equation
r = j - 2k + t(i + j + 3k)."

I have no clue with how to answer this question. I have considered on whether you would change the parametric vector equation into a Cartesian equation and somehow incorporate the point to find the distance from there although, it does sound a bit confusing to me. Moreover, how are you able to find the shortest distance? - As in, is there a particular approach to follow that allows you find the shortest distance from a point? It's something that I have not really come across in my course notes, lectures or textbook.

2. There ares several ways (including a direct formula) . One way:

A generic point of the line is

$\displaystyle R=(t,1+t,-2+3t)$

If $\displaystyle R$ is the point of the line to shortest distance from $\displaystyle P(-1,0,2)$ then,

$\displaystyle \vec{RP}\cdot (1,1,3)=0$ .

3. Find the equation of the plane containing the given point and having the given line as normal. Find the point at which the line intersects that plane. That is the point on the given line closest to the given point.

5. ## Re: Finding the shortest distance from a point to a line with parametric vector equat

Hi,
In several responses to this problem, a formula for the distance is indicated. I think the formula is complicated and not worth memorizing, but the derivation is so simple that you can recover the formula almost immediately. Here it is:

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# find the distance of the point from the line with parametric equation

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