Explain in detail the triangle law of addition.

1.	Explain in detail the triangle law of addition.
Answer» Solution :(i) 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. (iii) Let `vecR`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 `vecR`with vector `vecA`. (v) Thus we can write, `vecR = vecA + vecB` a) Magnitude of resultant vector (i) From `Delta ABN,` ` cos theta = (AN)/(B) , An = B cos theta"" [cos theta = (adj)/(hyp)]` ` sin theta = (BN)/(B) , BN = B sin theta "" [ sin theta = (OPP)/(hyp)]` (ii) From `DeltaOBN,`[PYTHOGORAS theoram `hyp^2 = adj^2 + opp^2`] ` OB^2= ON^2 + BN^2` `R^2 = (A+B cos theta)^2 + (B sin theta)^2` ` R^2 = A^2 + B^2 cos^2 theta + 2AB cos theta + B^2 sin^2 theta` ` R = \|vecA + vecB\| = sqrt(A^2 + B^2 + 2AB cos theta )` b) Direction of resultant vectors: From `DeltaΟΒΝ,` ` tan alpha = (BN)/(ON) = (BN)/(OA + AN)` ` tan alpha = ((Bsin theta)/(A + B cos theta))` ` alpha = tan^(-1) [(B sin theta)/(A + B cos theta)]`

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