Find the vector and Cartesian equations of the plane passing through the point (2, -1, 1) and perpendicular to the line having direction ratios 4, 2, -3.

Question

Philoid · Accepted Answer

Given :

A = (2, -1, 1)

Direction ratios of perpendicular vector = (4, 2, -3)

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) Dot Product :

If are two vectors

then,

3) Equation of plane :

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

Where,

For point A = (2, -1, 1), position vector is

Vector perpendicular to the plane with direction ratios (4, 2, -3) is

Now,

= 8 – 2 – 3

= 3

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

As

= 4x + 2y – 3z

Therefore, the equation of the plane is

4x + 2y – 3z = 3

Or

4x + 2y – 3z - 3 = 0