Obtain an expression for the magnitude and direction of the resultant of two concurrent and coplanar forces.

Explanation

To find the magnitude of the resultant :
form the `OAC,vecP+vecQ=vecR`
CD is `_|_` to OD. From the pythagoras theorem,
`OC^2=OD^2+DC^2`
but `DC^2=AC^2+AD^2`
From the right angled triangle ADC,
`sin theta=CD//AC`
`:. " "AC sin theta=CD`
`cos theta=AD//AC`
i.e., `AD=AC costheta`
Eqn. (1) Can be written as `OC^2=(OA+AD)^2+DC^2`
using (2) and (4) we get
`OC^2=OA^2+AC^2+20A.AC cor theta`
i.e., `R^2+P^2+Q^2+2PQ costheta`
Hence, `R=(P^2+Q^2+2PQcostheta)^(1//2)`
To find the direction of the resultant :
From the right angled triangle ODC,
`tanprop=(CD)/(OD)`
i.e., `prop=tan^(-1)(CD)/(OA+AD)`
`:." "prop=tan^(-1)[(ACsintheta)/(OA+ACcostheta)]` w.r.t `vecP`
i.e `prop=tan^(-1)[(Qsintheta)/(P+Qcostheta)]` w.r.t `vecP`