Given a plane
 |
(1)
|
and a point 
,
the normal vector to the plane
is given by
![v=[a; b; c],](/images/equations/Point-PlaneDistance/NumberedEquation2.svg) |
(2)
|
and a vector from the plane to the point is given by
![w=-[x-x_0; y-y_0; z-z_0].](/images/equations/Point-PlaneDistance/NumberedEquation3.svg) |
(3)
|
Projecting 
onto 
gives the distance 
from the point to the plane as
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(4)
|
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(5)
|
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(6)
|
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(7)
|
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(8)
|
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(9)
|
Dropping the absolute value signs gives the signed distance,
 |
(10)
|
which is positive if 
is on the same side of the plane as the normal vector 
and negative if it is on the opposite
side.
This can be expressed particularly conveniently for a plane specified in Hessian normal form by the simple equation
 |
(11)
|
where 
is the unit normal vector. Therefore, the distance of the plane from the origin is
simply given by 
(Gellert et al. 1989, p. 541).
Given three points 
for 
, 2, 3, compute the unit normal
 |
(12)
|
Then the (signed) distance from a point 
to the plane containing the three points is given by
 |
(13)
|
where 
is any of the three points. Expanding out the coordinates shows that
 |
(14)
|
as it must since all points are in the same plane, although this is far from obvious based on the above vector equation.
When the point lies in the plane determined by the other three points, it is said to be coplanar with them, and the distance given by the formulas above collapses to 0.
