Point of Intersection between a Line and a Plane

Point of Intersection between a Line passing through a point having position vector \(\vec{L_0}\) having a direction vector \(\vec{L}\) and a Plane passing through a point having position vector \(\vec{P_0}\) having a normal vector \(\vec{N}\) is given by following formula

\( \vec{P}= \vec{L_0} + (\frac{(\vec{P_0} - \vec{L_0}) \cdot \vec{N} }{\vec{L}\cdot\vec{N} }) \vec{L}\)

If \(\vec{L}\cdot\vec{N} \neq 0\), \(\vec{P}\) gives the position vector of point of intersection. If \(\vec{L}\cdot\vec{N} = 0\), then the line lies on the plane if \((\vec{P_0} - \vec{L_0}) \cdot \vec{N} =0 \) or is parallel to the plane if \((\vec{P_0} - \vec{L_0}) \cdot \vec{N} \neq 0 \).

Also If \(\vec{L}\cdot\vec{N} = 0\), the distance D between the Line and the Plane is given by the following formula

\( D = \frac{(\vec{P_0} - \vec{L_0}) \cdot \vec{N}}{\vert\vec{N}\vert}\)