Find the value of λ for which the four points with position vectors and are coplanar.

Given :

Let, A, B, C & D be four points with given position vectors

To Find : value of λ

Formulae :

1) Vectors :

If A & B are two points with position vectors ,

Where,

then vector is given by,

2) Scalar Triple Product:

If

Then,

3) Determinant :

Answer :

For given position vectors,

Vectors are given by,

………eq(1)

………eq(2)

………eq(3)

Now, for vectors

= -2(2 + 2λ) – 0 + 2(4)

= - 4 - 4λ + 8

= 4 – 4λ

………eq(4)

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

………eq(5)

From eq(4) and eq(5)

4 – 4λ = 0

4λ = 4