is parallel to a non-zero vector

is parallel to a non-zero vector (a) Parametric form of vector equation The parametric form of vector equation of the plane passing through two given distinct points A and B with position vectors a  and b , and parallel to a non-zero vector c  is r  = s b tc  or ( s a sb tc ... ( ) where , ,( s t ∈  and c  are not parallel vectors. (b) Non-parametric form of vector equation Equation ( ) can be written equivalently in non-parametric vector form as ) ((  = ... ( ) where (  and c  are not parallel vectors.

(c) Cartesian form of equation If x i y j z k b x i y j z k c i c j c k ≠ xi yj zk , then equation ( ) is equivalent to = This is the required Cartesian equation of the plane. Example . Find the non-parametric form of vector equation, and Cartesian equation of the plane passing through the point ( , , ) and parallel to the straight lines ) ( ) s ) t i We observe that the required plane is parallel to the vectors , k c = + passing through the point( , , ) with position vector a  . We observe that b is not parallel to  c .

Then the vector equation of the plane in non-parametric form is given by ( ) ( . …( ) Substituting and b  = in equation ( ), we get )) ( ⋅− = , which implies that ( ⋅− = . If xi yj zk is the position vector of an arbitrary point on the plane, then from the above equation, we get the Cartesian equation of the plane as or Example . Find the vector parametric, vector non-parametric and Cartesian form of the equation of the plane passing through the points ( , , ), ( , ) and parallel to the straight line The required plane is parallel to the given line and so it is parallel to the vector = + the plane passes through the points , = −+ Applications of Vector Algebra  vector equation of the plane in parametric form is s b tc  , where s , t ∈  which implies that s t i = −+ , where s , t ∈  .

 vector equation of the plane in non-parametric form is ( ) (( Now, = + we have )) ( ) −−+ ) ⇒  If xi yj zk is the position vector of an arbitrary point on the plane, then from the above equation, we get the Cartesian equation of the plane as