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Distance from a point to a plane.

  1. Jun 15, 2009 #1
    1. The problem statement, all variables and given/known data

    Find the distance of the point (2,4,7) from the plane determined by the points (3,4,4) , (4,5,7) and (6,8,9).

    2 common vectors for the plane

    [itex]\left(
    \begin{array}{c}
    1 \\
    1 \\
    3
    \end{array}
    \right)
    [/itex] [itex] \left(
    \begin{array}{c}
    3 \\
    4 \\
    5
    \end{array}
    \right) [/itex]

    The normal to the plane is then

    [itex]\left(
    \begin{array}{c}
    1 \\
    1 \\
    3
    \end{array}
    \right)\times \left(
    \begin{array}{c}
    3 \\
    4 \\
    5
    \end{array}
    \right)=\left(
    \begin{array}{c}
    -7 \\
    4 \\
    1
    \end{array}
    \right)[/itex]

    The vector from the point to a known point on the plane dot product with the normal vector divided by |n| gives

    [itex]\frac{\left(
    \begin{array}{c}
    1 \\
    0 \\
    -3
    \end{array}
    \right).\left(
    \begin{array}{c}
    -7 \\
    4 \\
    1
    \end{array}
    \right)}{\sqrt{7^2+4^2+1^2}}=\frac{10}{\sqrt{66}} = \frac{5\sqrt{66}}{33}[/itex]

    The answer has just 3 in the denominator, not 33. Have I done this correctly?
     
  2. jcsd
  3. Jun 15, 2009 #2

    Dick

    User Avatar
    Science Advisor
    Homework Helper

    That looks fine to me. May be a typo in the answers.
     
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