# Thread: distance from point to line

1. ## distance from point to line

This is the equation i have to slove.
Find the distance from the point (4, 4, 1) to the line

i know distance fromula from point to plane but i don't understand from point to line..

2. Originally Posted by DMDil
This is the equation i have to slove.
Find the distance from the point (4, 4, 1) to the line
The direction vector of the line is $D=<0,5,2>$.
The point $(0,4,1)$ is on the line.
Let $V=<4-0,4-4,1-1>=<4,0,0>$. The distance is $d = \frac{{\left\| {V \times D} \right\|}}{{\left\| D \right\|}}$

3. Thanks alot..i really understand now..

4. Since the shortest distance from a point to a line is along the perpendicular to the line, construct the plane through the given point perpendicular to the given line. The line is given by x= 0, y= 4+ 5t, z= 1+ 2t so the vector <0, 5, 2> is in the direction of the line and so normal to any perpendicular plane. That is, the plane perpendicular to that line and through the point (4, 4, 1) is 0(x- 4)+ 5(y- 4)+ 2(z-1)= 0 or 5y+ 2z= 22.
the line crosses that plane where 5(4+ 5t- 4)+ 2(1+ 2t- 1)= 22 or 29t= 22 so t= 22/29 and x= 0, y= 4+ 110/29= 226/29, z= 1+ 44/29= 73/29. The distance from (4,4,1) to (0, 225/29, 73/29) is distance from the point to the line. It should be just what you get using Plato's formula.