Q. 154.0( 4 Votes )

Find the value of λ for which the points A(3, 2, 1), B(4, λ, 5), C(4, 2, -2) and D(6, 5, -1) are coplanar.

Ans. λ = 5

Answer :

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




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