Explain in detail the triangle law of addition.

Explanation

Let vecA and `vecB` are two vectors they are inclined at angle `theta` between them.
(ii) According to triangle law of vector addition, head of the vector `vecA` is connected to tail of the vector `vecB` and both are represented in adjescent side of a triangle in some order.
Let `vecR` be the resultant vector, which is represented in third closing side of the represented in third closing side of the triangle in opposite order.
(iv) Let `alpha` be the angle made by the resultant vector `vecR` with vector `vecA`.
(v) Thus we can write, `vecR=vecA+vecB`

a) Mgnitude of resutant vector
(i) From `DeltaABN`,
`cos theta=(AN)/(B), AN=B cos theta" "[cos theta=(adj)/(hyp)]`
`sin theta=(BN)/(B), BN=B sin theta" "[sin theta=(opp)/(hpp)]`
(ii) From `DeltaOBN,` [Pythogoras theoram `hyp^(2)=adj^(2)+opp^(2)`]
`OB^(2)=ON^(2)+BN^(2)`
`R^(2)=(A+B cos theta)^(2)+(Bsin theta)^(2)+B^(2)sin^(2)theta`
`R=|vecA+vecB=sqrt(A^(2)+B^(2)+2AB cos theta)`
b) Direction of resultant vectors :
From `DeltaOBN`,
`tan alpha=(BN)/(ON)=(BN)/(OA+AN)`
`tan alpha=((B sin theta)/(A+B cos theta))`
`a=tan^(-1)[(B sin theta)/(A+B cos theta)]`