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(- λ – 10) – 3(13) + 1(28 – 5λ)

= 2λ + 20 – 39 + 28 – 5λ

= 9 – 3λ

………eq(4)

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

………eq(5)

From eq(4) and eq(5)

9 – 3λ = 0

3λ = 9