Using vector method, show that the points A(4, 5, 1), B(0, -1, -1), C(3, 9, 4) and
D(-4, 4, 4) are coplanar.

Given Points :

A ≡ (4, 5, 1)

B ≡ (0, -1, -1)

C ≡ (3, 9, 4)

D ≡ (-4, 4, 4)

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

Formulae :

4) Position Vectors :

If A is a point with co-ordinates (a₁, a₂, a₃)

then its position vector is given by,

5) Vectors :

If A & B are two points with position vectors ,

Where,

then vector is given by,

6) Scalar Triple Product:

If

Then,

7) Determinant :

Answer :

For given points,

A ≡ (4, 5, 1)

B ≡ (0, -1, -1)

C ≡ (3, 9, 4)

D ≡ (-4, 4, 4)

Position vectors of above points are,

Vectors are given by,

………eq(1)

………eq(2)

………eq(3)

Now, for vectors

= 4(15) – 6(21) + 2(33)

= 60 – 126 + 66

= 0

Hence, vectors are coplanar.

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

Note : Four points A, B, C & D are coplanar if and only if