# Distance from a point to a plane.

1. Jun 15, 2009

### Gregg

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

$\left( \begin{array}{c} 1 \\ 1 \\ 3 \end{array} \right)$ $\left( \begin{array}{c} 3 \\ 4 \\ 5 \end{array} \right)$

The normal to the plane is then

$\left( \begin{array}{c} 1 \\ 1 \\ 3 \end{array} \right)\times \left( \begin{array}{c} 3 \\ 4 \\ 5 \end{array} \right)=\left( \begin{array}{c} -7 \\ 4 \\ 1 \end{array} \right)$

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

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

The answer has just 3 in the denominator, not 33. Have I done this correctly?

2. Jun 15, 2009

### Dick

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

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