Question

a,b are two vectors than find resultant of vector a and b by parallelogram law of vectors. what is the formula for its direction. ​

Answer 1

Answer:

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Explanation:

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,

OB2=C2+BC2

or    OB2=(OA+AC)2+BC2. . . . . . ( i )

Intriangle ABC,

cosθ=ABAC

or ,      AC =  AB cosθ

or ,   AC = OD cosθ 

             = Q cosθ        [ AB =  OD = Q ]

Also, 

cosθ=ABBC

or ,    BC = AB sinθ

or ,   BC = OD sinθ 

             = Q sinθ                  [ AB = OD = Q }

Magnitude of resultant:

Substituting value of AC and BC in ( i ), we get

R2=(P+Qcosθ)2+(Qsinθ)2

or ,   R2=P2+2PQcosθ+Q2cos2θ+Q2sin2θ

or ,   R2=P2+2PQcosθ+Q2

R = P2+2PQcosθ+Q2

Which is the magnitude of resultant.

Direction of resultant :

Let ϕ be the angle made by resultant R with P . Then,

From triangle OBC,

tanϕ=OCBC=OA+ACBC

or ,