Relation between a Line and a Plane

1. A Line and a Plane in 3D Cartisean Space can be related in following ways
   1. The Line lies on the Plane
   2. The Line is Parallel to the Plane
   3. The Line Intersects the Plane at a Single Point
2. The Relation between a Line having a Direction Vector \(\vec{L}\) Passing through a Point having Position Vector \(\vec{L_0}\) and a Plane having a Normal Vector \(\vec{N}\) Passing through a Point having Position Vector \(\vec{P_0}\) can be found using following steps
   1. If \(\vec{L}\cdot\vec{N} = 0\), Calculate the Distance \(D\) between the Line and the Plane. If \(D \neq 0\) then the Line and the Plane are Parallel. Otherwise the Line lies on the Plane. This Distance can be calculated either by Finding the Distance of Point given by Vector \(\vec{L_0}\) from the Plane or by Finding the Distance of Point given by Vector \(\vec{P_0}\) from the Line.
   2. If \(\vec{L}\cdot\vec{N} \neq 0\) then the Line Intersects the Plane at a Single Point. The following gives the Derivation of formula for Finding Point of Intersection between the Line and the Plane
      
      Position Vector Equation for a Line having a Direction Vector \(\vec{L}\) Passing through a Point having Position Vector \(\vec{L_0}\) is given as
      
      \( \vec{P}= \vec{L_0} + t\vec{L}\)   ...(1)
      
      where \(t\) is Any Real Number
      
      Now, Non Parametric Vector Equation for a Plane having a Normal Vector \(\vec{N}\) Passing through a Point having Position Vector \(\vec{P_0}\) is given as
      
      \( (\vec{P}-\vec{P_0})\cdot\vec{N} =0\)   ...(2)
      
      where \(\vec{P}\) is the Position Vector of any Arbitrary Point on the Plane
      
      Since the Line and the Plane Intersect, putting the value of \(\vec{P}\) from equation (1) in equation (2) gives
      
      \( (\vec{L_0} + t\vec{L}-\vec{P_0})\cdot\vec{N} =0\)
      
      \(\Rightarrow t\vec{L}\cdot\vec{N} + (\vec{L_0} -\vec{P_0})\cdot\vec{N} =0\)
      
      \(\Rightarrow t\vec{L}\cdot\vec{N} = (\vec{P_0} -\vec{L_0})\cdot\vec{N} \)
      
      \(\Rightarrow t = \frac{(\vec{P_0} -\vec{L_0})\cdot\vec{N}}{\vec{L}\cdot\vec{N}}\)    ...(3)
      
      Putting the value of \(t\) from equation (3) in equation (1) we get the formula for Position Vector of Point of Intersection of the Line and the Plane as
      
      \( \vec{P}= \vec{L_0} + (\frac{(\vec{P_0} - \vec{L_0}) \cdot \vec{N} }{\vec{L}\cdot\vec{N} }) \vec{L}\)    ...(4)