Find the equation of the plane passing through the point (2, 3, 1) given that the direction ratios of normal to the plane are proportional to 5, 3, 2.

Given: The plane is passing through P(2, 3, 1) and perpendicular to the line having 5, 3, 2 as the direction ratios.

To find: the equation of the plane

Let the position vector of this point P be

And it is also given the plane is normal having 5, 3, 2 as the direction ratios.

Then

We know that the vector equation of a plane passing through the point and perpendicular/normal to the vector is given by

Substituting the values from eqn(i) and eqn(ii) in the above equation, we get

(by multiplying the two vectors using the formula )

is the vector equation of a required plane.

Let

Then, the above vector equation of the plane becomes,

Now multiplying the two vectors using the formula, we get

This is the Cartesian form of the equation of the required plane.