A plane can be completely illustrated by denoting two intersecting lines which can be translated into a fixed point A and two nonparallel direction vectors. The position vector of any general point P on the plane passing through point A and having direction vectors and is given by the equation
Vector equation of a plane
Parametric equation of a plane: λ , μ are called a parameters λ,μ
If N is considered to be normal to a given plane, then all other normals to that plane are considered parallel to N which are resultantly scalar multiples of N., In particular,we can say that there are two normals of length 1:
Normal/Scalar product form of vector equation of a plane
Consider a vector n passing through a point A. Only one plane through A can be is perpendicular to the vector. Now consider R being any point on the plane other than A as shown above. Then we can say that
Cartesian equation of a plane
Therefore, the Cartesian form is
where n1, n2 and n3 are the components of n and where n is called the normal vector.
Example: Find the equation of the plane passing through the three points P1(1,-1,4), P2(2,7,-1), and P3(5,0,-1).
Hence, consider one point on the plane:
In vector form:
Any non-zero scalar multiples of is also a normal vector of the plane. Therefore, Multiply by -1.
Example: Find the equation of the plane with normal vector containing the point (-2, 3, 4).
Example:Find the distance of the plane = 8 from the origin, and the unit vector perpendicular to the plane.
Example: Find the Cartesian equation of the plane through the point A (1, 1, 1) perpendicular to the vector
Example: Show that the following vector is perpendicular to the plane containing the points A(1, 0, 2), B(2, 3, -1) and C(2, 2, -1 ).
In conclusion, n is a vector that is perpendicular to 2 vectors in the plane so is perpendicular to the plane.