Vector Addition
One resultant from two forces: the parallelogram rule
Summary
When you add two vectors, the resultant vector (R) is the diagonal of the parallelogram they form. You add the components separately: Rₓ = Aₓ + Bₓ, Rᵧ = Aᵧ + Bᵧ. The magnitude comes from Pythagoras, the angle from atan2.
In depth
Vectors carry both a magnitude and a direction, so they do not add like ordinary numbers. Given two vectors A and B, their sum R is the resultant vector that starts at the same point and reaches the opposite corner of the parallelogram. Split into components, algebraically: Rₓ = Aₓ + Bₓ and Rᵧ = Aᵧ + Bᵧ. The magnitude is |R| = √(Rₓ² + Rᵧ²) and the direction is atan2(Rᵧ, Rₓ). This simple principle underlies countless applications: adding forces in physics (Newton), combining course and wind in navigation, combining field vectors in electromagnetism, adding up character movements in game engines, and even combining gradients in machine learning.
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