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JEE MainMathematicsThree Dimensional Geometry

Let OP and OQ be two vectors such that | OP | = 4 , | OQ | = 5 , and OP OQ = 16 . A point R divides the line segment PQ internally in the ratio :1 ( 0 ). If the projection of OR on OP is equal to the magnitude of the cross product of OR and OQ , then the value of is :

Options

  1. A1 2
  2. B4
  3. C2
  4. D1 4

Correct answer

C. 2

Step-by-step solution

By the section formula, the position vector of R is: OR = OQ + OP + 1 The projection of OR on OP is given by OR OP | OP | . First, find OR OP : OR OP = ( OQ OP ) + | OP |^2 + 1 Given OP OQ = 16 and | OP | = 4 , we have | OP |^2 = 16 . OR OP = 16 + 16 + 1 = 16( + 1) + 1 = 16 The projection is 16 4 = 4 . Next, find the magnitude of the cross product of OR and OQ : | OR OQ | = | OQ + OP + 1 OQ | Since OQ OQ = 0 , this simplifies to: | OR OQ | = | OP OQ | + 1 Using the identity | OP OQ |^2 + ( OP OQ )^2 = | OP |^2 | OQ

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