Using paralalogram law of vector addition to derive an expression for the magnitude and direction of the resultant of two vector?
Parallelogram Law of Vector Addition
Statement of Parallelogram Law
If two vectors acting simultaneously at a point can be represented both in magnitude and direction by the adjacent sides of a parallelogram drawn from a point, then the resultant vector is represented both in magnitude and direction by the diagonal of the parallelogram passing through that point.
Derivation of the law
Note: All the letters in bold represent vectors and normal letters represent magnitude only.
Let P and Q be two vectors acting simultaneously at a point and represented both in magnitude and direction by two adjacent sides OA and OD of a parallelogram OABD as shown in figure.
Let θ be the angle between P and Q and R be the resultant vector. Then, according to parallelogram law of vector addition, diagonal OB represents the resultant of P and Q.
So, we have
R = P + Q
Now, expand A to C and draw BC perpendicular to OC.
From triangle OCB,
In triangle ABC,
Magnitude of resultant:
Substituting value of AC and BC in (i), we get
which is the magnitude of resultant.
Direction of resultant: Let ø be the angle made by resultant R with P. Then,
From triangle OBC,
which is the direction of resultant.
Hope you understand..