Consider two vectors 
as shown in the figure. Vector 
is displaced parallel to itself till the tail end of both the vectors touch at a point O. The parallelogram is completed as shown in the figure. Applying the law of triangle of vectors, to the triangle OAC, we have,
The above example can be stated in the following way as the law of parallelogram of vectors - If two vectors, acting simultaneously at a point, can be represented both in magnitude and direction by the two adjacent sides of a parallelogram drawn from a point, then the resultant is represented completely, both in magnitude and direction by the diagonal of the parallelogram passing through the point.
In the figure, two vectors are completely represented by the two sides OA and OB respectively of a parallelogram. Then, according to the law of parallelogram of vectors, the diagonal OC of the parallelogram will be resultant 
, such that 
.
Let q be the angle between two given vectors . From c, drop a perpendicular CN on OA produced. In the right angled D ANC,
Also,
ON = OA + AN = P + Q Cos q
Consider the right angled DONC,
The direction of the resultant is given by the angle b which is determined in the following way
Special cases
In this case, q = 00
b = 00 i.e., the resultant vector also points in the direction of the given vector.
In this case,
The resultant acts in the direction of the larger vector
