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Given a vector, how to compute orthogonal plane

  1. Oct 9, 2015 #1


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    Given a vector (in 3-d), how do I determine the plane that is orthogonal to it?

    I am not quite finding a search term that gets me to this, but instead to several similar, but different questions.

    One such is find an equation of a plane perpendicular to a vector and passing through a given point. I may have been staring at this too long, but I am not visualizing why the given point is needed.

  2. jcsd
  3. Oct 9, 2015 #2
    When the plane is given by
    [tex] a x + b y + c z + d = 0, [/tex]
    the normal vector of that plane is [itex] (a,b,c). [/itex] Then, you just replace the normal vector to the vector what you want and find [itex] d [/itex] by using any point on the plane.
  4. Oct 9, 2015 #3
    There are infinitely many planes orthogonal to a given vector, so you would also need to specify a point on the plane to calculate its equation. You can write a plane with normal [itex]\bf{n}[/itex] in vector notation as the set of all [itex]\bf{x} [/itex] such that [tex] \bf{x \: . n} = d [/tex] Where [itex] d[/itex] is a scalar determined by a given point. Alternatively you can expand this out in cartesian coordinates as Daeho Ro wrote above.
  5. Oct 12, 2015 #4


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    The plane would go through one end of the vector.

    For example, a 3-d vector would go from point (0,0,0) to (x,y,z). I want the plane perpendicular to this vector, going through (x,y,z). Do I need another point to define the plane, or is this enough information.

  6. Oct 12, 2015 #5


    Staff: Mentor

    To get the equation of a plane, you need a normal to the plane, and two points on the plane, one of which is arbitrary and has coordinates (x, y, z). Call these points P(x, y, z) and ##P_0(x_0, y_0, z_0)##, where the coordinates ##x_0, y_0, z_0## are known.
    Form a direction vector ##\vec{v} = <x - x_0, y - y_0, z - z_0>##. The normal and this vector are perpendicular, so their dot product ##\vec{v} \cdot \vec{N} = 0##. This dot product will produce the equation of the plane.
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