. . Equation of a plane passing through a given point and parallel to two given non-parallel vectors. (a) Parametric form of vector equation Consider a plane passing through a given point A with position vector a and parallel to two given non-parallel vectors b and c .
If r is the position vector of an arbitrary point P on the plane, then the vectors ( ), a b and c are coplanar. So, ( lies in the plane containing b and c . Then, there exists scalars , s t ∈ such that r sb tc which implies r = a sb tc , where , s t ∈ ... ( ) This is the parametric form of vector equation of the plane passing through a given point and parallel to two given non-parallel vectors .
(b) Non-parametric form of vector equation Equation ( ) can be equivalently written as ) ( = ... ( ) which is the non-parametric form of vector equation of the plane passing through a given point and parallel to two given non-parallel vectors . (c) Cartesian form of equation If x i y j z k b b i b j b k c i c j c k xi yj zk , then the equation ( ) is equivalent to = This is the Cartesian equation of the plane passing through a given point and parallel to two given non-parallel vectors.