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
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
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