Question

state triangle law of vector addition. find the magnitude and direction of resultant ​

Answer 1

Answer:

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

Explanation:

To obtain R⃗  which is the resultant of the sum of vectors A⃗  and B⃗  with the same order of magnitude and direction as shown in the figure, we use the following rule:

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 (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.