. . Equation of a plane passing through three given non-collinear points (a) Parametric form of vector equation Theorem . If three non-collinear points with position vectors , , a b c are given, then the vector equation of the plane passing through the given points in parametric form is s b t c , where , ≠ ≠ and , s t ∈ .
Proof Consider a plane passing through three non-collinear points , , A B C with position vectors , , a b c respectively. Then atleast two of them are non-zero vectors. Let us take b ≠ and c ≠ . Let r be the position vector of an arbitrary point P on the plane.
Take a point D on AB (produced) such that AD is parallel to AB and DP is parallel to AC . Therefore, AD = ( ), s b DP t c . Now, in triangle ADP , we have AP = AD DP or s b t c , where , ≠ ≠ and , s t ∈ . That is, r = s b t c .
This is the parametric form of vector equation of the plane passing through the given three non-collinear points. (b) Non-parametric form of vector equation Let , , A B and C be the three non collinear points on the plane with position vectors , , a b c respectively. Then atleast two of them are non-zero vectors. Let us take b ≠ c ≠ .
Now AB and AC . The vectors ( and ( lie on the plane. Since , , a b c are non-collinear, AB is not parallel to AC . Therefore, ) ( is perpendicular to the plane.
If r is the position vector of an arbitrary point ( , , ) P x y z on the plane, then the equation of the plane passing through the point A with position vector a and perpendicular to the vector ( ) ( is given by ) (( ) ( )) = or [ ] a b a c This is the non-parametric form of vector equation of the plane passing through three non-collinear points. (c) Cartesian form of equation If ( , ),( x y z y z are the coordinates of three non-collinear points , , A B C with position vectors , , a b c respectively and ( , , ) x y z is the coordinates of the point P with position vector r , then we have x i y j z k b x i y j z k x i y j z k xi yj zk Fig. . Fig.
. O B C P O D B P C Vector - - Applications of Vector Algebra Using these vectors, the non-parametric form of vector equation of the plane passing through the given three non-collinear points can be equivalently written as = which is the Cartesian equation of the plane passing through three non-collinear points.