Question

•Question :-

→ Derive traingle law of vector
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Answer 1

Answer:

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 .

Derivation of triangle law of vector addition

From triangle OCB,

OB2=OC2+BC2

OB2=(OA+AC)2+BC2 (eq.1)

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 in (eqn.1), we get

R2=(P+QcosΘ)2+(QsinΘ)2

R2=P2+2PQcosΘ+Q2cos2Θ+Q2sin2Θ

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=QsinΘP+QcosΘ

therefore, ϕ=tan−1(QsinΘP+QcosΘ)

Above equation in the direction of the resultant vector.

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Answer 2

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Triangle law of vector addition states that when two vectors are represented as two sides of the triangle with the order of magnitude and direction, then the third side of the triangle represents the magnitude and direction of the resultant vector.