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Points on lines with parametric equations (linear algebra)

  1. Mar 3, 2015 #1
    1. The problem statement, all variables and given/known data
    "Let L1 be the line having parametric equations : x = 2 - s, y = -1 + 2s, z = 1+s and L2 be the line:
    x = 1 +t, y = 2+ t, z =2t .

    a. Do the lines intersect? If so, find the point of intersection.

    b. Find the point P on the graph of L1 that is closest to the graph of L2 and find the point Q on the graph of L2 that is closest to the graph of L1. Hint: Use the fact that the vector PQ will be orthogonal to the direction vectors of both lines. "

    2. Relevant equations


    3. The attempt at a solution
    In part a, I set the parametric equations equal to each other and solved for t and s. It looks like the lines do not intersect.

    I'm not sure how to go about part b. How does the hint that the vector PQ will be orthogonal to the direction vectors help me?
    The direction vectors would be:
    L1 = (-1, 2, 1) L2 = (1, 1, 2)

    Any help is greatly appreciated!
     
    Last edited: Mar 3, 2015
  2. jcsd
  3. Mar 3, 2015 #2

    Mark44

    Staff: Mentor

    Should the x-coordinate of L2 be x = 1 + 2t?
     
  4. Mar 3, 2015 #3
    Ahh sorry, It should be x = 1+ t
     
  5. Mar 3, 2015 #4

    Mark44

    Staff: Mentor

    At the points on the lines that are closest to each other, the segment joining the two lines will be perpendicular to each line.
     
  6. Mar 3, 2015 #5
    Okay, will it have something to do with PQ ⋅ L1 = 0 and PQ ⋅L2 = 0? Or am I way off here?
     
  7. Mar 3, 2015 #6

    LCKurtz

    User Avatar
    Science Advisor
    Homework Helper
    Gold Member

    Mark's hint is different from the hint given in the original post. Take a general point on each line, one depends on ##s## and the other on ##t##. Write the distance (easier to use distance^2) between those two points. Minimize that using calculus methods to minimize a function of two variables. The ##s## and ##t## you get will give you the two points you seek.
     
  8. Mar 3, 2015 #7
    What would a general point on the parametric lines look like? The x, y, z components give in the problem?
    ( I haven't taken multi variable calc yet and this class assumes I will only use algebra to complete the homework)
     
  9. Mar 3, 2015 #8

    Mark44

    Staff: Mentor

    I don't see how what I said was different. In the OP, it states that "Hint: Use the fact that the vector PQ will be orthogonal to the direction vectors of both lines."
    A completely different approach uses the hint above by taking the cross product of the two direction vectors. Then, take two arbitrary points, one from each line, and create a displacement vector between those two points. The shortest distance between the two lines will have to be a scalar multiple of the vector you obtained from the cross product.
     
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