The plane is passing through the point (1, -1, 1). Let the position vector of this point be

And it is also given the plane is normal to the line joining the points A(1, 2, 5) and B(-1, 3, 1).

Then

Position vector of - position vector of

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 )

Multiplying by (-1) on both sides we get,

is the vector equation of a plane passing through the point (1, -1, 1) and normal to the line joining the point (1, 2, 5) and (-1, 3, 1).

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 equation of a plane passing through the point (1, -1, 1) and normal to the line joining the point (1, 2, 5) and (-1, 3, 1).