I know how to find the distance between a point and a line, not between two lines.
Find the shortest distance between the lines $(-1,1,4) + t(1,1,-1)$ and $(5,3,-3) + s(-2,0,1)$
Any help would be appreciated.
Linear Algebra – Finding the Shortest Distance Between Two Lines
linear algebra
Best Answer
The distance between two lines in $ \Bbb R^3 $ is equal to the distance between parallel planes that contain these lines.
To find that distance first find the normal vector of those planes - it is the cross product of directional vectors of the given lines. For the normal vector of the form (A, B, C) equations representing the planes are:
$ Ax + By + Cz + D_1 = 0 $
$ Ax + By + Cz + D_2 = 0 $
Take coordinates of a point lying on the first line and solve for D1.
Similarly for the second line and D2.
The distance we're looking for is: $$d = \frac{|D_1 - D_2|}{\sqrt{A^2 + B^2 + C^2}}$$