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JEE Main20238 Apr 2023Evening ShiftMathematicsVector AlgebraActual

Let the vectors u 1 → = i ^ + j ^ + a k ^ , u 2 → = i ^ + b j ^ + k ^ , and u 3 → = c i ^ + j ^ + k ^ be coplanar. If the vectors v 1 → = a + b i ^ + c j ^ + c k ^ , v 2 → = a i ^ + b + c j ^ + a k ^ and v → 3 = b i ^ + b j ^ + c + a k ^ are also coplanar, then 6 a + b + c is equal to

Options

  1. A0
  2. B4
  3. C12
  4. D6

Correct answer

C. 12

Step-by-step solution

Given: u → 1 = i ^ + j ^ + a k ^ ,   u → 2 = i ^ + b j ^ + k ^ and u → 3 = c i ^ + j ^ + k ^ are coplanar. Also given that, v → 1 = a + b i ^ + c j ^ + c k ^ ,   v → 2 = a i ^ + b + c j ^ + a k ^ and v → 3 = b i ^ + b j ^ + c + a k ^ are coplanar. Now, using the condition of coplanar we get, ⇒ 1 1 a 1 b 1 c 1 1 = 0 Expanding the determinant along R 1 . ⇒ b - 1 - 1 - c + a 1 - b c = 0 ⇒ a + b + c = 2 + a b c           . . . . .

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