If O is the origin and P(1, 2, -3) be a given point, then find the equation of the plane passing through P and perpendicular to OP.

Given :

P = (1, 2, -3)

O = (0, 0, 0)

To Find : Equation of a plane

Formulae :

1) Position vectors :

If A is a point having co-ordinates (a₁, a₂, a₃), then its position vector is given by,

2) Vector :

If A and B be two points with position vectors respectively, where

then,

3) Dot Product :

If are two vectors

then,

4) Equation of plane :

If a plane is passing through point A, then the equation of a plane is

Where,

For points,

P = (1, 2, -3)

O = (0, 0, 0)

Position vectors are

Vector

Now,

= 1 + 4 + 9

= 14

And

= x + 2y + 3z

Equation of the plane passing through point A and perpendicular to the vector is

But,

Therefore, the equation of the plane is

x + 2y + 3z = 14

x + 2y + 3z – 14 = 0