derivation of R is equal to root of a square + b square + 2 a b cos theta
Answers
Step-by-step explanation:
Triangle Law of Vector Addition Derivation
Consider two vectors A and B that are represented in the order of magnitude and direction by the sides OA and AB respectively of the triangle OAB. Let R be the resultant of vectors P and Q.
Derivation of triangle law of vector addition
R=A+B
From triangle OCB,
OB2=OC2+BC2 OB2=(OA+AC)2+BC2 (eq.1)
In triangle ACB with ϴ as the angle between A and B
cosΘ=ACAB AC=ABcosΘ=BcosΘ sinΘ=BCAB BC=ABsinΘ=BsinΘ R2=(A+BcosΘ)2+(BsinΘ)2 (after substituting AC and BC in eq.1)
R2=A2+2ABcosΘ+B2cos2Θ+B2sin2Θ R2=P2+2PQcosΘ+Q2
therefore, R=√P2+2PQcosΘ+Q2−−−−−−−−−−−−−−−−−
Above equation is the magnitude of the resultant vector.
To determine the direction of the resultant vector, let ɸ be the angle between the resultant vector R and P.
From triangle OBC,
tanϕ=BCOC=BCOA+AC tanϕ=BsinΘP+BcosΘ
therefore, ϕ=tan−1(BsinΘA+BcosΘ)
Above equation is the direction of the resultant vector.