If O is the origin and the coordinates of A are (a, b, c). Find the direction cosines of OA and the equation of the plane through A at right angles to OA.

As it is given that the required plane is passing through A(a, b, c) and is perpendicular to OA. Let the position vector of this point A be

And it is also given the plane is normal to the line joining the points O(0,0,0) and A(a, b, c)

Then

Position vector of - position vector of

Therefore the direction ratios of OA are proportional to a, b, c

Hence the direction cosines are

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 equation of the required plane.