Given :

Points A, B, C & D are coplanar where,

A ≡ (3, 2, 1)

B ≡ (4, λ, 5)

C ≡ (4, 2, -2)

D ≡ (6, 5, -1)

To Find : value of λ

Formulae :

1) Position Vectors :

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

then its position vector is given by,

2) Vectors :

If A & B are two points with position vectors ,

Where,

then vector is given by,

3) Scalar Triple Product:

If

Then,

4) Determinant :

Answer :

For given points,

A ≡ (3, 2, 1)

B ≡ (4, λ, 5)

C ≡ (4, 2, -2)

D ≡ (6, 5, -1)

Position vectors of above points are,

Vectors are given by,

………eq(1)

………eq(2)

………eq(3)

Now, for vectors

= - 1(9) – (2 - λ).(7) – 4(3)

= - 9 – 14 + 7λ – 12

= 7λ – 35

………… eq(4)

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

………… eq(5)

From eq(4) and eq(5)

7λ – 35 = 0