# Distance between a point and a line in space

I have two points $P_1(x_1, y_1, z_1)$ and $P_2(x_2, y_2, z_2)$ on a line, $L$, and another point $P_0(x_0, y_0, z_0)$.

I want to find the distance between $P_0$ and $L$. Could someone help?

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Hmm... –  Ｊ. Ｍ. Dec 22 '10 at 14:25
why at (9) there is only x? what about y and z? –  nkint Dec 22 '10 at 14:31
$\mathbf{x}$ is a vector, it has all three components. –  Ross Millikan Dec 22 '10 at 14:37
and | x_2 - x_1 | is vector magnitude or absolute value? –  nkint Dec 22 '10 at 14:54
Vector magnitude. –  Raskolnikov Dec 22 '10 at 15:14

The distance $h$ from the point $P_0=(x_0,y_0,z_0)$ to the line passing through $P_1=(x_1,y_1,z_1)$ and $P_2=(x_2,y_2,z_2)$ is given by $h=2A/r$, where $A$ is the area of a triangle defined by the three points and $r$ is the distance from $P_1$ to $P_2$. The values of $r$ and $A$ can be computed as follows:

$r=\sqrt{(x_2-x_1)^2+(y_2-y_1)^2+(z_2-z_1)^2}$ and $A=\frac{1}{2}\sqrt{a_1^2+a_2^2+a_3^2},$

where

$a_1=x_0y_1+x_1y_2+x_2y_0 - (y_0x_1+y_1x_2+y_2x_0),\\ a_2=y_0z_1+y_1z_2+y_2z_0 - (z_0y_1+z_1y_2+z_2y_0),\\ a_3=x_0z_1+x_1z_2+x_2z_0 - (z_0x_1+z_1x_2+z_2x_0).$

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I thought to furnish a picture:

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Please cite your source of this image as it is clear that you did not create it. –  Daniel Rust Feb 18 at 13:00

look here: http://paulbourke.net/geometry/pointline/ All you need to do is extend it to 3d.

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The link is dead. –  Andrey Sokolov Jun 20 '13 at 1:41

The shortest distance from a point to a line is always perpendicular to the given line. Here, the given line is in the direction of the vector $\langle x_2-x_1, y_2-y_1, z_2-z_1\rangle$. The plane $(x_2- x_1)(x- x_0)+ (y_2- y_1)(y- y_0)+ (z_2- z_1)(z- z_0)= 0$ has that vector as normal vector and contains the point $(x_0, y_0, z_0)$ so the shortest distance is through that plane. Determine where the given line intersects that plane. The shortest distance is the distance form that point of intersection to $(x_0, y_0, z_0)$.

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