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    OB’  (vector Q) and OA (vector P) are to be added and they are drawn from O.  Let the angles they make with the X-axis be ф2 and ф1 respectively.  Draw the vector AB = Q = OB from the arrow of vector P, parallel to OB’.

    Now draw OB (vector R).  We have to prove  vector OA + vector AB = vector OB.
Draw perpendiculars AF, AG from A and perpendiculars BE , BH from B onto the X and Y axes respectively.

Component of vector (OA) P along X-axis = OF = P Cos ф1
 Component of vector (AB)Q along X-axis = -FE = -Q Cos (180- ф2) = Q Cos ф2
 Net resultant component along X-axis = P Cos ф1 + Q Cos ф2 = OF – FE = OE.

   Y component of vector P or OA = AF = GO
   Y component of vector Q or AB = GH
  Net resultant along the Y-axis = P Sin ф1 + Q Sin ф2 = GO+GH = OH

  X-component of vector OB = OE  = same as that for vector OA + AB.
  Y-Component of vector OB = OH = same as for the vector OA + AB
 Vectors     OA + AB = vector   OB 
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