Question

1. If a and 6 are the two vectors with an angle'e' between them, then to find resultant the steps to be followed are 1) arrange the two vectors coinitially as the adjacent sides of a parallelogram II) arrange the two vectors coterminally as the adjacent sides of a parallelogram III) draw a diagonal which passes through their common point IV) draw a diagonal which passes through any point​

Answer 1

Explanation:





Join / Login

Class 11

>>Physics

>>Motion in a Plane

>>Mathematical Operations on Vectors

>>Give the derivation for resultant of two

Question



Give the derivation for resultant of two vectors and for the angle between resultant and a vector.

Hard

Open in App

Solution



Verified by Toppr

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