Answer :

Given Points :


A ≡ (4, 5, 1)


B ≡ (0, -1, -1)


C ≡ (3, 9, 4)


D ≡ (-4, 4, 4)


To Prove : Points A, B, C & D are coplanar.


Formulae :


4) Position Vectors :


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


then its position vector is given by,



5) Vectors :


If A & B are two points with position vectors ,


Where,




then vector is given by,




6) Scalar Triple Product:


If





Then,



7) Determinant :



Answer :


For given points,


A ≡ (4, 5, 1)


B ≡ (0, -1, -1)


C ≡ (3, 9, 4)


D ≡ (-4, 4, 4)


Position vectors of above points are,






Vectors are given by,




………eq(1)




………eq(2)




………eq(3)


Now, for vectors







= 4(15) – 6(21) + 2(33)


= 60 – 126 + 66


= 0



Hence, vectors are coplanar.


Therefore, points A, B, C & D are coplanar.


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


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