Find the shortest distance between two lines:

In summary, to find the shortest distance between two skew lines, you can find the cross product of the directions of the two lines, and then take the projection of the vector between two points on the lines in the direction of the cross product vector.
  • #1
Simkate
26
0
Given the following two skew lines:
L1: (0, 4, -3) + s(-1, 1, 3)
L2: (1, 2, 5) + t(-3, 2, 5)

Find the shortest distance.



MY WORK::
Cross-product of the lines (-1, 1, 3) X (-3, 2, 5) = (-1, -4, 1) with length 3*sqrt(2)
Vector between the points (0, 4, -3) - (1, 2, 5) = (-1, 2, -8)

Dot product of those results (-1, 2, -8) . (-1, -4, 1) = -15

Remove sign and divide by cross-product length 15 / (3 sqrt(2)) = 5/sqrt(2)




I was wondering if the VECTOR between the points is right or is the reverse
(1,2,5)-(0,4,-3)= (1, -2, 13) ?
 
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  • #2
Simkate said:
Given the following two skew lines:
L1: (0, 4, -3) + s(-1, 1, 3)
L2: (1, 2, 5) + t(-3, 2, 5)

Find the shortest distance.



MY WORK::
Cross-product of the lines (-1, 1, 3) X (-3, 2, 5) = (-1, -4, 1) with length 3*sqrt(2)
Vector between the points (0, 4, -3) - (1, 2, 5) = (-1, 2, -8)

Dot product of those results (-1, 2, -8) . (-1, -4, 1) = -15

Remove sign and divide by cross-product length 15 / (3 sqrt(2)) = 5/sqrt(2)




I was wondering if the VECTOR between the points is right or is the reverse
(1,2,5)-(0,4,-3)= (1, -2, 13) ?

The two lines can be thought of as lying in two parallel planes. The cross product of the directions of the two lines gives you a vector that is perpendicular to these planes.

Your first vector between the two points on the lines, <-1, 2, -8> is correct, but the opposite, <1, -2, 8> would also work. Was <1, -2, 13> a typo?

What you want is the projection of the vector <1, -2, 8> in the direction of the vector <-1, 4, 1> (I didn't check your cross-product work). That will give you the shortest distance between the two lines.
 

What is the formula for finding the shortest distance between two lines?

The formula for finding the shortest distance between two lines is the distance formula in three dimensions.

Can the shortest distance between two lines be negative?

No, the shortest distance between two lines cannot be negative. It is always a positive value.

Do the two lines have to be parallel to find the shortest distance between them?

No, the two lines do not have to be parallel to find the shortest distance between them. They can be intersecting or skew lines.

Can the shortest distance between two lines be greater than the distance between any two points on the lines?

Yes, it is possible for the shortest distance between two lines to be greater than the distance between any two points on the lines. This can occur if the two lines are skew or non-parallel.

Is there a specific scenario where finding the shortest distance between two lines is useful?

Finding the shortest distance between two lines can be useful in various fields such as engineering, physics, and computer graphics. It can help determine the closest approach of two objects or the shortest distance a moving object can travel to avoid a fixed obstacle.

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