Triangle Law of Vector Addition Derivation
Consider two vectors A and B representing the two sides of the triangle OP and PQ respectively. Let vector R (OQ) be the resultant vector of the addition of A and B.
According to the above description, we draw the below diagram.
From the triangle OSQ,
OQ2 = OS2 + QS2
OQ2 = (OP +PS)2 + QS2 ——(1)
In triangle PSQ with θ as the angle between A and B
cos θ = PS / PQ
PS = PQ cosθ = B cosθ
sin θ = QS / PQ
QS = PQ sinθ = B sinθ
Substituting the values of PS and QS in equation (1), we get
R2 = (A + Bcosθ)2 + (Bsinθ)2
R2 = A2 + 2ABcosθ + B2cos2θ + B2sin2θ
R2 = A2 + 2ABcosθ + B2
Therefore,
R = √(A2+ B2 + 2ABcosθ)
The above equation represents the magnitude of resultant vector.
To find the direction of the resultant vector R, let Φ be the angle between vectors A and R.
From triangle, OQS
tanΦ = QS / OS
tanΦ = QS / (OP + PS)
tanΦ = Bsinθ / (A + Bcosθ)
therefore,
Φ = tan-1[Bsinθ / (A + Bcosθ)]
The above equation gives the direction of the resultant vector R.
Read More
Triangle Law of Vector Addition
The Triangle Law of Vector Addition is a method used to add two vectors. It states that when two vectors are represented as two sides of a triangle in sequence, the third side of the triangle is taken in the opposite direction. It represents the resultant vector in both magnitude and direction.
Vectors are the backbone of many technologies nowadays, such as computer graphics, visual effects, machine learning, and artificial intelligence. Therefore, understanding the addition of vectors is a much-needed skill to understand these further advanced topics.
Let’s learn more about Triangle Law of Vector Addition in detail with steps to add two vectors with formula below.
Contact Us