Distance from a Plane

Distance Between a Point and a Plane

Distance D of a point C from any plane having normal vector \(\vec{A}\) making a vector \(\vec{B}\) with any point on the plane is given by following formula

\(D = \frac{\vec{A}\cdot\vec{B}}{\vert \vec{A} \vert} \hspace{.5cm} D_{abs} = \frac{|\vec{A}\cdot\vec{B}|}{\vert \vec{A} \vert}\)

The above formula gives a signed distance values (i.e. the distance values can be negative or positive). To find the actual distance absolute value must be taken.

If a plane is given by equation \(ax + by + cz +d\) then distance D of this plane from a point C having coordinates (x1,y1,z1) is given by following formula

\(D = \frac{ax_1 + by_1 + cz_1 + d}{\sqrt{a^2 + b^2 + c^2}} \hspace{.5cm} D_{abs} = \frac{|ax_1 + by_1 + cz_1 + d|}{\sqrt{a^2 + b^2 + c^2}}\)

The above formula also gives a signed distance values (i.e. the distance values can be negative or positive). To find the actual distance absolute value must be taken.
The distance from the origin is given by the following

\(D = \frac{d}{\sqrt{a^2 + b^2 + c^2}} \hspace{.5cm} D_{abs} = \frac{|d|}{\sqrt{a^2 + b^2 + c^2}}\)

Distance Between a Line and a Plane

Following are the steps to find out Distance Between a Line having direction vector \(\vec{A}\) and a Plane having normal \(\vec{B}\)

  1. Find \(\vec{A}\cdot\vec{B}\).
  2. If the value of the dot product is not zero then the line intersects plane and distance does not have any meaning.
  3. If the value of the dot product is zero then the distance of the line from the plane can be found out by finding the distance of any point on line from plane by any of the methods given above.

Distance Between 2 Planes

Following are the steps to find out Distance Between a Planes having normal \(\vec{A}\) and \(\vec{B}\)

  1. Find \(\vec{A}\times\vec{B}\).
  2. If the value of the cross product is not zero then the planes intersect and distance does not have any meaning.
  3. If the value of the cross product is zero then the distance between the planes can be found out by finding the distance of any point on one plane to the other plane by any of the methods given above. Distance between 2 parallel planes can also be found out by finding the absolute value of difference of the distance of the planes from origin.