Proof (Higher Dimensions)
The general formula for higher dimensions can be quickly arrived at using vector notation. Let the hyperplane have equation, where the is a normal vector and is a position vector to a point in the hyperplane. We desire the orthogonal distance to the point . The hyperplane may also be represented by the scalar equation, for constants . Likewise, a corresponding may be represented as . The magnitude of the vector is like our distance above, so we desire the scalar projection in the direction of . Noting that (as satisfies the equation of the hyperplane) we have
- .
Notice how the general expression is consistent with dimensions.
Read more about this topic: Perpendicular Distance
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