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

enter image description here

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    $\begingroup$ Please cite your source of this image as it is clear that you did not create it. $\endgroup$ – Dan Rust Feb 18 '14 at 13:00

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