How Do You Solve a Skew Lines Distance Problem in Vector Geometry?

[thepassenger48]

Homework Statement

Given
Line1: (x,y,x)=(2,1,-1) + t(-1,0,2)
Line2: (x,y,x)=(1,1,3) + s(5,-6,-7)

The lines L1 and L2 are skew; determine points Q and R on lines L1 and L2 respectively such that ||QR|| = 2

2. The attempt at a solution
I have no idea how to even approach this question! Can someone give me a clue? Please

Thank you

 

[Dick]

Do you know how to find the closest distance between the two lines? I attempted to do it and get sqrt(16/21) which is less than 2. This means, if you imagine the geometry, that there are immensely many points on the two lines that are distance 2 apart. Are there any two in particular that you are interested in?

 

[Architeuthis Dux]

for your question. This is a tough vector geometry problem, and it will require some careful thinking and calculations to solve. Here are some steps you can follow to approach this problem:

1. Understand what skew lines are: Skew lines are two lines that do not intersect and are not parallel. This means that they are not on the same plane and do not have the same direction.

2. Understand the equations of the lines: The equations of the lines are given in parametric form, which means that they can be written as a point plus a scalar multiple of a direction vector. For example, line L1 can be written as (2,1,-1) + t(-1,0,2). This means that for any value of t, you can find a point on this line by plugging in t and solving for the coordinates.

3. Use the distance formula: The problem asks for points Q and R on lines L1 and L2, respectively, such that the distance between them is 2. This means that you can use the distance formula to set up an equation to solve for the values of t and s.

4. Set up the equation: Using the distance formula, you can set up an equation with the coordinates of points Q and R, and the direction vectors of lines L1 and L2. You can then solve for t and s using algebraic methods.

5. Find the points: Once you have solved for t and s, you can plug these values back into the equations of lines L1 and L2 to find the coordinates of points Q and R.

I hope this helps you get started on solving this problem. Remember to carefully consider the definitions and equations involved, and don't be afraid to ask for help if you get stuck. Good luck!