Show that the four points A, B, C and D with position vectors and respectively are coplanar.

OR

The scalar product of the vector with a unit vector along the sum of vectors and is equal to one. Find the value of λ and hence find the unit vector along

The position vectors of points

, , and

The points A, B, C and D are coplanar if the vectors , and are coplanar

Vectors , and are coplanar of

Let us first write the vectors , and

Now let us find the value of

represents the determinant

Expand the determinant along the first row

Hence vectors , and are coplanar and hence points A, B, C and D are coplanar

OR

, and

Given that the dot product of with unit vector along is 1

To find a unit vector along we have to divide by

Hence unit vector along

Take dot product of this unit vector along with

Square both sides,

⇒ (2 + λ)² + 40 = (6 + λ)²

⇒ 40 = (6 + λ)² – (2 + λ)²

⇒ 40 = (6 + λ + 2 + λ)(6 + λ – 2 – λ)

⇒ 40 = (8 + 2λ)4

⇒ 10 = 2(4 + λ)

⇒ 5 = 4 + λ

⇒ λ = 1

Hence λ = 1.