Find the vector, and Cartesian equations of a plane which passes through the point (1, 4, 6) and the normal vector to the plane is

Given :

A = (1, 4, 6)

To Find : Equation of plane.

Formulae :

1) Position Vector :

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

2) Dot Product :

If are two vectors

then,

3) Equation of plane :

Equation of plane passing through point A and having as a unit vector normal to it is

Where,

Position vector of point A = (1, 4, 6) is

Now,

= (1×1) + (4×(-2)) + (6×1)

= 1 – 8 + 6

= - 1

Equation of plane is

This is vector equation of the plane.

As

Therefore

= (x × 1) + (y × (-2)) + (z × 1)

= x - 2y + z

Therefore equation of the plane is

This is Cartesian equation of the plane.