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Let the position vectors of points P , Q , R and S be a → = i ^ + 2 j ^ - 5 k ^ , b → = 3 i ^ + 6 j ^ + 3 k ^ , c → = 17 5 i ^ + 16 5 j ^ + 7 k ^ and d → = 2 i ^ + j ^ + k ^ , respectively. Then which of the following statements is true?

Options

  1. AThe points P ,   Q ,   R and S are NOT coplanar
  2. Bb → + 2 d → 3 is the position vector of a point which divides P R internally in the ratio 5 : &#16
  3. Cb → + 2 d → 3 is the position vector of a point which divides P R externally in the ratio 5 : &#16
  4. DThe square of magnitude of the vector b → × d → is 95

Correct answer

B. b → + 2 d → 3 is the position vector of a point which divides P R internally in the ratio 5 : &#16

Step-by-step solution

Given, P ( 1 , 2 , - 5 ) ,   Q ( 3 , 6 , 3 ) ,   R 17 5 , 16 5 , 7   &   S ( 2 , 1 , 1 ) Now solving options, Option A checking points are coplanar, P Q → = 2 , 4 , 8 ,   P R → = 12 5 , 6 5 , 12   &   P Q → = 1 , - 1 , 6 Now if points are coplanar then, P Q → P R → P S → = 0 Now solving L.H.S, we get, 2 4 8 12 5 6 5 12 1 - 1 6 = 2 5 1 2 4 12 6 60 1 - 1 6 = 2 5 96 - 24 - 72 = 0 Hence, they are coplanar So option A is wrong. Now solving, o

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