1. Limited time only! Sign up for a free 30min personal tutor trial with Chegg Tutors
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Equation of a line through a point and another line at 90 degrees.

  1. Nov 19, 2011 #1
    1. The problem statement, all variables and given/known data
    Find the equation of the line through the point (-5,-4,2) that intersects the line at r = (7,-13,8) +t(1,2,-2) at 90 degrees.

    2. Relevant equations
    I guess that using the dot product for the direction vector of the line needed, (x,y,z) , with the direction of the given line will = 0. Then i get lost as to what I should be doing afterwards, in incorporating the point given with the variables that result from the dot product.

    3. The attempt at a solution

    So i did (x,y,z) dot (1,2,-2) to get x +2y -2z = 0
    at this point, i'm not sure where to do.

    Btw, the answer is r = (-5,-4,2) +t(14,-5,2).
     
  2. jcsd
  3. Nov 19, 2011 #2

    LCKurtz

    User Avatar
    Science Advisor
    Homework Helper
    Gold Member

    Your whole problem will be solved if you can figure out the direction from Q that is perpendicular to your line. Call P = (7,-13,8) and Q = (-5,-4,2) and V the vector from P to Q. Call the direction vector of your line D= <1,2,-2>. You know
    W= VxD is perpendicular to both V and D.

    Now think about what direction WxD would be.
     
  4. Nov 19, 2011 #3

    HallsofIvy

    User Avatar
    Staff Emeritus
    Science Advisor

    I would do this in a slightly different way. The given line has "direction vector" <1, 2, -2>. A line that passes throught (-5, -4, 2) and is perpendicular to the given line must lie in the plane having that vector as normal vector and containing that point: that plane is 1(x+ 5)+ 2(y+ 4)- 2(z- 2)= 0. Find the point at which the given line intersects that plane and use that and (-5, -5, 2) to determine the line.
     
  5. Nov 19, 2011 #4
    I tried both ways, but i'm getting slightly different answers. Could you please show me some clear steps because I'm getting somewhat confused.
     
  6. Nov 19, 2011 #5
    A note to take into consideration is that we haven't really touched upon 3D intersections, using perpendicular vectors. Never.
     
  7. Nov 19, 2011 #6

    LCKurtz

    User Avatar
    Science Advisor
    Homework Helper
    Gold Member

    If you tried my suggestion and made no arithmetic errors, you can shorten your final vector by dividing it by 9 to get your suggested direction. Was that your problem?
     
  8. Nov 19, 2011 #7
    Well, I got (16,-7,-3) as my final answer.. You got the actual answer? Could you show your steps please?
     
  9. Nov 19, 2011 #8

    LCKurtz

    User Avatar
    Science Advisor
    Homework Helper
    Gold Member

    V = Q-P = <-12,9,-6>
    W = V x D = <-6,-30,-33>
    Normal = W x D = <126,-45,18>

    Divide by 9 to shorten it: <14,-5,2>. The numbers would likely be smaller if I had divided V by 3 to begin with.
     
  10. Nov 19, 2011 #9
    Ahh, thank you so much. I realized that I made a mistake half way through, but i changed the wrong number in the last cross. Thanks! By the way, if you don't mind, why did you choose to complete the operations that you did. My teacher only gives very basic explanations, if any, and sometimes I do get confused.
     
  11. Nov 19, 2011 #10

    LCKurtz

    User Avatar
    Science Advisor
    Homework Helper
    Gold Member

    There are several ways to work that problem. You could have found the vector projection of V = PQ on your line and subtracted that from V to get the direction. You could use the method HallsOfIvy suggested. Or you could use what I suggested. They all work.

    The reason the method I showed you works is that W = V x D is perpendicular to the plane of V and D, so when you cross W with D, the result, being perpendicular to W must be back parallel to the plane of V and D. And since it is perpendicular to D, it is the required direction. It might be helpful for you to draw a picture.
     
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook




Similar Discussions: Equation of a line through a point and another line at 90 degrees.
Loading...