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JEE Main20254 Apr 2025Evening ShiftMathematicsVector AlgebraActual

Let the three sides of a triangle A B C be given by the vectors 2 i - j + k , i -3 j -5 k and 3 i -4 j -4 k . Let G be the centroid of the triangle A B C . Then 6 (| AG |^2+| BG |^2+| CG |^2 ) is equal to ________

Correct answer

0

Step-by-step solution

By given data AB + AC = CB Let pv of A are O then AB = B - A i.e. p v of B -=2 i - j + k CA = A - C i.e. pv of C =-( i -3 j -5 k ) Now pv of centroid aligned & ( G )-= A + B + C 3 = 0 +(2,-1,1)+(-1,3,5) 3 & G = 1 3 ( i +2 j +6 k ) aligned Now AG = 1 3 ( i +2 j +6 k ) | AG |^2= 1 9 41 BG = ( 1 3 -2 ) i + ( 2 3 +1 ) j +(2-1) k | BG |^2= 59 9 aligned & CG = ( 1 3 +1 ) i + ( 2 3 -3 ) j +(2-5) k & | CG |^2= 146 9 aligned Now aligned & 6 [| AG |^2+| BG |^2+| CG |^2 ]=6 [ 41 9 + 59 9 + 146 9 ] & =6 246 9 =164 aligned

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