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Math Help - Distance between planes (Proof)

  1. #1
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    Distance between planes (Proof)

    Consider two plans P1: ax + by + cz + e = 0 and P2: ax + by +cz + f = 0. Show that the distance between P1 and P2 is given by d = l e-f l / sqrt(a^2 + b^2 +c^2)

    * any idea on what to do.
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  2. #2
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    Quote Originally Posted by sderosa518 View Post
    Consider two plans P1: ax + by + cz + e = 0 and P2: ax + by +cz + f = 0. Show that the distance between P1 and P2 is given by d = l e-f l / sqrt(a^2 + b^2 +c^2)
    * any idea on what to do.
    Look at reply #9 on this thread.
    Because the planes are parallel, use any point on either plane find the distance to the other plane.
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  3. #3
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    here is my solution:

    l Ap + Bq + Cr + D l / sqrt(A^2 + B^2 + C^2)

    point: (p, q, r)

    n = P1: Ax + By + Cz + e = 0 and P2: Ax + By + Cz + f = 0

    Isolate e and f and get:

    e = - (Ax1 + By2 + Cz3)

    - f = Ax1 + By2 + Cz3

    Distaance between Plane 1 and 2 have the same exact distance as (p1, q1, r1) due to the fact its on the same plane.

    l Ap + Bq + Cr + D l / sqrt(A^2 + B^2 + C^2)

    D = l -f + e l / sqrt(A^2 + B^2 + C^2)

    D = l e - f l / sqrt(A^2 + B^2 + C^2)
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  4. #4
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    Good work. It is correct.
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