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MHT CET202527 Apr 2025Evening ShiftMathematicsVector AlgebraActual

If 4 i +7 j +8 k , 2 i +3 j +4 k and 2 i +5 j +7 k are the position vectors of the vertices A , B and C , respectively, of triangle ABC , then the position vector of the point in which bisector of B meets C A is

Options

  1. Ai (4 13 +12)+ j (7 13 +30)+ k (8 3 +42)
  2. Bi (4 13 +12)+ j (7 13 +30)+ k (8 13 +42) 13 -6
  3. Ci (4 13 +12)+ j (7 13 +30)+ k (8 13 +42) 13 +6
  4. Di (4 13 +12)+ j (7 13 +30)- k (8 13 +42) 6- 13

Correct answer

A. i (4 13 +12)+ j (7 13 +30)+ k (8 3 +42)

Step-by-step solution

Given: a = 4 i + 7 j + 8 k b = 2 i + 3 j + 4 k c = 2 i + 5 j + 7 k Vectors from vertex B: BA = a - b = 2 i + 4 j + 4 k BC = c - b = 2 j + 3 k Magnitudes: | BA | = 2^2 + 4^2 + 4^2 = 6 | BC | = 2^2 + 3^2 = 13 By the Angle Bisector Theorem, point D divides AC in the ratio | BA | : | BC | = 6 : 13 . Applying the section formula: d = 13 a + 6 c 13 + 6 Substituting the given vectors: d = (4 13 + 12) i + (7 13 + 30) j + (8 13 + 42) k 13 + 6 This expression corresponds to option C.

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