state triangle law of vector addition give its analytical treatment
Answers
Triangle Law of Vector Addition
Consider two vectors P⃗ and Q⃗ that are represented in the order of magnitude and direction by the sides OA and AB, respectively of the triangle OAB. Let R⃗ be the resultant of vectors P⃗ and Q⃗ .
From triangle OCB,
OB2=OC2+BC2
OB2=(OA+AC)2+BC2
In triangle ACB with ϴ as the angle between P and Q
cosΘ=ACAB
AC=ABcosΘ=QcosΘ
sinΘ=BCAB
BC=ABsinΘ=QsinΘ
Substituting the values of AC and BC .we get
R2=(P+QcosΘ)2+(QsinΘ)2
R2=P2+2PQcosΘ+Q2cos2Θ+Q2sin2Θ
R2=P2+2PQcosΘ+Q2
therefore, R=✓P^2+2PQcosΘ+Q^2
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=QsinΘP+QcosΘ
therefore,
ϕ=tan−1(QsinΘP+QcosΘ)
Above equation in the direction of the resultant vector.
