Distance between two lines in R^3

[SteveDC]

Homework Statement

Let line L1 be the line through origin O and the point B(2,-1,-2). Let line L2 be the line through point A(3,1,-1) with direction vector i-j-k. Obtain a vector n, orthogonal to both lines. Use n and the vector OA connecting the lines to find the shortest distance between two lines.

Homework Equations

The Attempt at a Solution

I have used both direction vectors for both lines to find their cross product to obtain the vector n, orthogonal to both lines: (2i-j-2k) x (i-j-k) = -i-k

But then not sure where to go from here despite the hint in the question of using vector OA and n?

 

[Simon Bridge]

If you drew a line-segment joining the closest points on the lines to each other, what sort of properties would it have?

 

[SteveDC]

It would have direction n and have two points, one lying on L1 and one lying on L2, but I can't think of a way of finding these points?

 

[D H]

Given some point p₁ on line 1 and some point p₂ on line 2, what is the projection of the vector from point p₁ to point p₂ onto the vector n that is normal to both lines?

 

[SteveDC]

Am I right in saying that vector p1>p2 projected onto n would always be the distance between the two lines, but even if this is the case, how would you find the smallest value for this to find the shortest distance.

 

[D H]

Write an expression for some arbitrary point p₁ on line 1 and some arbitrary point p₂ on line 2. What is the projection of p₂-p₁ onto a vector perpendicular to both lines?

It's perhaps better if you any two arbitrary lines rather than your specific lines because the result is very general.

Hint: Use ##\vec p_i + \lambda\vec \tau_i## as the equation of a line passing through some point p_i.

 

[HallsofIvy]

The line of "shortest distance" between two lines is, as said above, perpendicular to both lines. Find the plane perpendicular to the first line at an arbitrary point [itex](x_0, y_0, z_0)[/itex]. Determine the point, [itex](x_1, y_1, z_1)[/itex], where the second line crosses that plane. Determine what [itex](x_0, y_0, z_0)[/itex] must be so the line between [itex](x_0, y_0, z_0)[/itex] and [itex](x_1, y_1, z_1)[/itex] is also pependicular the second line.

 

[D H]

That's an alternative approach, Halls, that isn't quite pertinent to the problem at hand. The OP was asked to use the normal to both lines and a line segment connecting a point on one line to a point on the other line.