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Distance of line to origin

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ehild

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1. Homework Statement



Say you have an equation of a line in 3 space and you want to find the distance from the line to the origin how do you go about it?
I guess by definition the distance of a line to a point ( say the origin in this case) is the minimum distance such that the line and some vector are perpendicular. That kind of sucks how I said it.

So x is a unique point on the line so you want the distance xo ( where o is the origin) such that xo is perpendicular to the line L . Not really sure how to do this. It is in the cross product section so maybe that would work. I tried to use dot product but I don't know how to get the shortest point. If I did I should dot xo with L and it would be 0. Well that was my thinking. I don't really know how to go about this.
Thanks
You have the line given with a point (position vector r0) and direction vector u:

r=r0 +tu (t a parameter).

You want the distance of the line from the origin. It is the length of the segment of a line which goes through the origin and crosses the original line perpendicularly. You need to find the point P on the line, so as vector d and vector u are perpendicular.

Make a plane normal to the line and going through the origin. You get the equation of that plane from the condition that all vectors r lying in it are normal to u : ru=0

Find the vector d that is in the plane and also on the line:

d=r0 +tu,*
du=0**

Multiply * with u and make it equal to zero. Isolate t, the parameter corresponding to point P. Substitute t back into *: you get the vector d. Determine the magnitude: it is the distance of the line from the origin.
 

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Bacle2

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Just to add to what has been said: think of the orthogonal projection of the line onto the zero subspace. This is what I think others are suggesting.
 
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thanks
 

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