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

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