Explain in detail the triangle law of addition.

Explanation

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

1) Magnitude of resultant vector
(i) From `DeltaABN`,
`costheta=(AN)/B, AN=Bcostheta" "[ cos theta=(adj)/(hyp)]`
`sintheta=(BN)/B, BN=B sintheta" "[sintheta=(opp)/(hyp)]`
(ii) From `DeltaOBN`, [Pythogoras theorem `hyp^(2)=adj^(2)+opp^(2)]`
`OB^(2)=ON^(2)+BN^(2)`
`R^(2)=(A+Bcostheta)^(2)+(Bsintheta)^(2)`
`R^(2)=A^(2)+B^(2)cos^(2)theta+2ABcostheta+B^(2)sin^(2)theta`
`R=abs(vec(A)+vec(B))=sqrt(A^(2)+B^(2)+2ABcostheta)`
2) Direction of resultant vectors:
From `DeltaOBN`,
`tan alpha=(BN)/(ON)=(BN)/(OA+AN)`
`tan alpha=((Bsintheta)/(A+Bcostheta))`
`" "alpha=tan^(-1)[(Bsintheta)/(A+Bcostheta)]`