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JEE MainMathematicsVector Algebra

Let H and O be the orthocentre and the circumcentre of a ABC , respectively. Then the vector sum HA + HB + HC is equal to

Options

  1. A2 OH
  2. BHO
  3. COH
  4. D2 HO

Correct answer

D. 2 HO

Step-by-step solution

Let the circumcentre O be the origin of the reference frame. The position vectors of the vertices A , B , and C are OA , OB , and OC respectively. We know that the position vector of the orthocentre H relative to the circumcentre O is given by OH = OA + OB + OC . The given expression is HA + HB + HC . Using the triangle law of vector addition, we can write HA = OA - OH , and similarly for the others. Thus, HA + HB + HC = ( OA - OH ) + ( OB - OH ) + ( OC - OH ) = ( OA + OB + OC ) - 3 OH Substituting OA + OB + OC = O

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