# Shortest distance between two skew lines

• iamsmooth
In summary, the shortest distance between the two lines L1 and L2 is 6.87 units. This was found by using the distance formula and solving for the distance between two points on each line. The logic for the algorithm was initially mixed up, but it was figured out by getting the parametric equation from the two points and solving from there.
iamsmooth

## Homework Statement

What is the shortest distance between these two lines?

L1:(x,y,z)=(4,−2,−2)+t(1,1,−3)
L2: The line through the points (−2,−2,0) and (−4,−5,0)

distance formula

## The Attempt at a Solution

I thought I was on the right track but apparently not.

For L1, I took 2 arbitrary t's to get 2 points on the line (which looking back, I think might be a wrong way to approach)

With t=1 and t=3, I got the points (5,-1,-5) for t=1 and (7,1,-11) for t=3.
P2-P1 = (2,2,-6) for L1

Now since the points for L2 are given:
P2-P1 = (-2,-3,0) for L2

Now I can take the cross product of the two lines:

$$\left| \begin{array}{ccc} i & j & k \\ 2 & 2 & -6 \\ -2 & -3 & 0 \end{array} \right| = -18i -12j -2k$$Plugging this all into the distance formula equation I get:

$$\frac{-4(-18)-5(-12)+6(-2)}{\sqrt{-18^2-12^2-2^2}} = \frac{120}{\sqrt{472}}$$

However, the answer is wrong. Any idea what I did wrong?

Thanks

For one thing, if I take the cross the two direction vectors of your lines, I get (-18,12,-2). And for another thing, where did you get the (-4,-5,6) you plugged into the distance formula?

I totally mixed up the logic for the algorithm, but I figured it out. I got the parametric equation from the 2 points, and from there solved it. Sorry for the trouble.

## What is the definition of "shortest distance between two skew lines"?

The shortest distance between two skew lines is the shortest length between any two points on the two lines that are perpendicular to each other.

## How do you find the shortest distance between two skew lines?

To find the shortest distance between two skew lines, you can use the formula d = |(P1-P2) · (n1 x n2)| / |n1 x n2|, where P1 and P2 are any two points on the lines, and n1 and n2 are the direction vectors of the two lines.

## Can the shortest distance between two skew lines be negative?

No, the shortest distance between two skew lines cannot be negative. It is always a positive value as it represents a physical length.

## What happens if the two skew lines are parallel?

If the two skew lines are parallel, then there is no shortest distance between them as they never intersect. In this case, the shortest distance is considered to be infinity.

## What is the significance of finding the shortest distance between two skew lines?

The shortest distance between two skew lines is a useful mathematical concept in various fields such as engineering, physics, and computer graphics. It can help determine the closest distance between two objects or points in 3-dimensional space and is essential in solving many real-world problems.

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