# 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

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