what is the triangle law of vector addition

The triangle law of vector addition states that if two vectors are represented by two sides of a triangle taken in order, then their sum (the resultant vector) is represented by the third side of the triangle taken in the opposite direction. Essentially, when the tail of one vector is joined to the head of the other, the vector from the tail of the first to the head of the second is the resultant vector. To apply it:

• Place the tail of the second vector at the head of the first vector.
• The resultant vector is drawn from the tail of the first vector to the head of the second vector.
• This forms a triangle, where the third side represents both the magnitude and direction of the resultant vector.

The magnitude ∣R∣|R|∣R∣ of the resultant vector R⃗=A⃗+B⃗\vec{R}=\vec{A}+\vec{B}R=A+B is given by the formula:

∣R∣=A2+B2+2ABcos⁡θ|R|=\sqrt{A^2+B^2+2AB\cos \theta}∣R∣=A2+B2+2ABcosθ​

where AAA and BBB are the magnitudes of the two vectors and θ\theta θ is the angle between them. The direction ϕ\phi ϕ of the resultant vector relative to the first vector is:

ϕ=tan⁡−1(Bsin⁡θA+Bcos⁡θ)\phi =\tan^{-1}\left(\frac{B\sin \theta}{A+B\cos \theta}\right)ϕ=tan−1(A+BcosθBsinθ​)

This law is fundamental in vector addition and is widely used in physics and engineering to find resultant displacement, forces, velocities, etc. The triangle law provides a geometrical method commonly known as the head-to-tail method of vector addition.