1. Not finding help here? Sign up for a free 30min 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!

Distance from a point to a plane

  1. Aug 28, 2016 #1
    1. The problem statement, all variables and given/known data
    What is the distance from the point P to the plane S?

    2. Relevant equations
    ## S = \left \{ r_{0} + s(u_{1},u_{2},u_{3})+t(v_{1},v_{2},v_{3}) | s,t \in \mathbb{R} \right \} ##

    3. The attempt at a solution

    I found an expression for the general distance between point P and a point on S, then found an expression for the distance and took the partial derivatives , ## \frac{\partial r}{\partial s} ## and ## \frac{\partial r}{\partial t}## (both of the argument of the square root function) and set both to zero, then solved the resulting linear equations. Is this the right method?
     
  2. jcsd
  3. Aug 28, 2016 #2

    Orodruin

    User Avatar
    Staff Emeritus
    Science Advisor
    Homework Helper
    Gold Member

    It will work, but there is an easier way to do it by finding a normal vector of the plane and projecting the difference vector between the point and any point in the plane on it.
     
  4. Aug 28, 2016 #3
    Wouldn't that just give me another vector in terms of x, y and z that I'd have to minimize?
     
  5. Aug 28, 2016 #4

    Orodruin

    User Avatar
    Staff Emeritus
    Science Advisor
    Homework Helper
    Gold Member

    No. It gives you the difference vector between the point and the closest point in the plane. You get the distance by computing its magnitude. (Note the projection part of the procedure!)
     
  6. Aug 28, 2016 #5
    Got it. Thanks for the help.
     
  7. Aug 28, 2016 #6

    LCKurtz

    User Avatar
    Science Advisor
    Homework Helper
    Gold Member

    Equivalently, if ##\vec V## is a vector from P to any point Q on the plane, and ##\hat n## is a unit normal to the plane, then$$|\vec V \times \hat n| = |\vec V|\cdot 1 \cdot \sin\theta = d$$so all you have to remember is to take the magnitude of ##\vec V \times \hat n##.
    Edit, added: Please ignore this post. This method works for finding the distance from a point to a line, where ##\hat n## is a unit direction vector along the line.
     
    Last edited: Aug 29, 2016
  8. Aug 29, 2016 #7

    Orodruin

    User Avatar
    Staff Emeritus
    Science Advisor
    Homework Helper
    Gold Member

    This is not equivalent. You want ##\vec V\cdot \hat n##. The cross product you quote is also dependent on which point in the plane you chose.
     
  9. Aug 29, 2016 #8

    LCKurtz

    User Avatar
    Science Advisor
    Homework Helper
    Gold Member

    You're correct of course. I must have posted that before I was fully awake.
     
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook

Have something to add?
Draft saved Draft deleted



Similar Discussions: Distance from a point to a plane
Loading...