Show that the fou

Given Points :

A = (3, 2, -5)

B = (-1, 4, -3)

C = (-3, 8, -5)

D = (-3, 2, 1)

To Prove : Points A, B, C & D are coplanar.

To Find : Equation of plane passing through points A, B, C & D.

Formulae :

1) Position vectors :

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

2) Equation of line

If A and B are two points having position vectors then equation of line passing through two points is given by,

3) Cross Product :

If are two vectors

then,

4) Dot Product :

If are two vectors

then,

5) Coplanarity of two lines :

If two lines are coplanar then

6) Equation of plane :

If two lines are coplanar then equation of the plane containing them is

Where,

For given points,

A = (3, 2, -5)

B = (-1, 4, -3)

C = (-3, 8, -5)

D = (-3, 2, 1)

Position vectors are given by,

Equation of line passing through points A & B is

Let,

Where,

&

And the equation of the line passing through points C & D is

Let,

Where,

&

Now,

Therefore,

= 72 + 48 – 120

= 0

……… eq(1)

And

= - 72 + 192 – 120

= 0

……… eq(2)

From eq(1) and eq(2)

Hence lines are coplanar

Therefore, points A, B, C & D are also coplanar.

As lines are coplanar therefore equation of the plane passing through two lines containing four given points is

Now,

= 24x + 24y + 24z

From eq(1)

Therefore, equation of required plane is

24x + 24y + 24z = 0

x + y + z = 0