Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Parametric Description of a Plane

  1. May 7, 2014 #1
    I read the definition that a plane is a point and two vectors with the equation being plane sum = {OP + tv + sw} where v and w are vectors and t and s are real numbers. This is called the parametric description of the plane. I haven't seen the equation in this form before though.

    Can someone explain what these values stand for/how to use this equation or direct me to a page that explains it? I just see the regular plane equation when I google this.
     
  2. jcsd
  3. May 7, 2014 #2

    Simon Bridge

    User Avatar
    Science Advisor
    Homework Helper
    Gold Member
    2016 Award

    You are aware that a plane can be defined by three points that are not co-linear?
    This form is just the same - using two pairs of points to form the two vectors.

    Any two vectors ##\vec v## and ##\vec w## must lie in a common plane.
    In fact, they can define a set of parallel planes.

    The parametric equation is just the instructions to get to another point in the plane starting from where you are at.

    i.e. If point ##P## is in the plane defined by the above vectors, then you can get from there to point ##Q##, also in the pane, by starting out at ##P## and walking ##t## steps of size ##v## in the direction of ##\vec v##, then turning to the direction of ##\vec w## and walking ##s## steps of size ##w## in that direction.

    In maths that is: ##Q=P+t\vec v + s\vec w##

    Concrete example:$$\begin{pmatrix}x\\y\\z\end{pmatrix}=\begin{pmatrix}1\\-1\\2\end{pmatrix}+t\begin{pmatrix}3\\4\\0\end{pmatrix}+s\begin{pmatrix}-1\\0\\0\end{pmatrix}$$... tells you how to get to point (x,y,z) from point (1,-1,1) using (3,4,0) and (-1,0,0) as cardinal directions.

    i.e. you want to get to (x,y)=(8,8), then t=2 and s=-1
    so you must travel 10 units along the hypotenuse of the 3-4-5 triangle, then 1 unit parallel to the x-axis.

    Notice that the plane in the example is parallel to the cartesian x-y plane, and have expressed the vectors in cartesian coordinates. I don't have to.

    The directions done this way basically translate the (t,s) coordinates for positions on the plane to (x,y,z) cartesian coordinates ...
     
    Last edited: May 7, 2014
  4. May 7, 2014 #3
    Thanks. That cleared it up. I had never thought of Q like that, but it really helped.
     
  5. May 8, 2014 #4

    Simon Bridge

    User Avatar
    Science Advisor
    Homework Helper
    Gold Member
    2016 Award

    No worries.
    Similarly the parametric equation for a line is ##Q=P+s\vec v## and for a 3D volume you need three parameters.
    It gets more fun when you use surfaces instead of planes - those are allowed to curve.
     
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook




Similar Discussions: Parametric Description of a Plane
  1. Parametrizing a curve (Replies: 4)

  2. Parametric equations (Replies: 4)

Loading...