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Physics Aleksandr
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Derive an expression for magnitude and direction of resultant vector using parallelogram law​

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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.

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