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JEE Main20264 April 2026Evening ShiftMathematicsVector AlgebraActual

Let u and v be unit vectors inclined at an acute angle such that | u v |= 3 2 . If A = u + v +( u v ) , then is equal to:

Options

  1. A4 3 ( A u )- 2 3 ( A v )
  2. B2 3 ( A u )- 1 3 ( A v )
  3. C4 3 ( A u )+ 2 3 ( A v )
  4. D( A u )- 1 2 ( A v )

Correct answer

A. 4 3 ( A u )- 2 3 ( A v )

Step-by-step solution

Given | u v | = 3 2 and u , v are unit vectors. = 3 2 Since the angle is acute, = 3 . u v = 3 = 1 2 Given A = u + v + ( u v ) . Taking the dot product with u : A u = ( u u ) + v u + ( u v ) u A u = (1) + 1 2 + 0 = + 1 2 Taking the dot product with v : A v = ( u v ) + v v + ( u v ) v A v = ( 1 2 ) + 1 + 0 = 2 + 1 From the first equation, we have 1 = 2( A u ) - 2 . Substituting this into the second equation: A v = 2 + 2( A u ) - 2 A v = 2( A u ) - 3 2 3 2 = 2( A u ) - ( A v ) = 4 3 ( A u ) - 2 3 ( A v ) Answer: 4 3 (

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