A plane passes through the point (1, -2, 5) and is perpendicular to the line joining the origin to the point . Find the vector and Cartesian forms of the equation of the plane.

As per the given criteria the required plane is passing through Q (1, -2, 5) and is perpendicular to OP, where point O is the origin and position vector of point P is . Let the position vector of this point Q be

And it is also given the plane is normal to the line joining the points O(0,0,0) and position vector of point P is

Then

Position vector of - position vector of

We know that vector equation of a plane passing through 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 equation of the required plane.