Distance of a point from a plane
Kapitoly: Distance of a point from a line, Distance of a point from a plane, Distance of two straight lines
The distance of a point from a plane is equal to the size of the "shortest" line segment drawn from that point to the plane.
Assignment
We have a plane given by the general equation r: 3x + 1y − 2z + 4 = 0 and a point X[5, 7, 2]. We ask what is the distance of the point X from the plane r. We would plot it in the figure as follows:
In the figure, we have a plane ABCD and a point X. The size of the line segment XF then determines the distance of the point X from the plane. This line is again perpendicular to the plane, so it is the shortest distance of the point to the plane. (The figure does not show the plane r: 3x + 1y − 2z + 4 = 0, nor the point X[5, 7, 2], just the distance of the point from the plane in general.)
Solution using the formula
We can use pretty much the same formula as when we calculated the distance of a point to a line, just add another dimension. So we calculate the distance of the point X[x1, x2, x3] and the plane r: ax + by + cz + d = 0 as
$$ v(X, r) = \frac{|a\cdot x_1+b\cdot x_2+c\cdot x_3+d|}{\sqrt{a^2+b^2+c^2}} $$