Find the vector and Cartesian equations of the planes
(a) that passes through the point and the normal to the plane is
(b) that passes through the point and the normal vector to the plane is

Solution:

(a) That passes through the point and the normal to the plane is
Let’s say that the position vector of the point be

It is known that Normal to the plane is given as:

Vector eq. of the plane is given as:

Now,

The required Cartesian equation of the plane is

Since,

So eq.(1) becomes,

As a result, the required Cartesian equation of the plane is

(b) That passes through the point and the normal vector to the plane is
Let’s say that the position vector of the point be

It is known that Normal to the plane is given as:

Vector eq. of the plane is given as:

Now,

Since,

So eq.(1) becomes,


As a result, the required Cartesian equation of the plane is