Dot Product of Two 2D Vectors (Components)

The dot product pairs up matching components, multiplies, and adds, and the answer is a plain number, not a vector. That single scalar measures agreement of direction: positive when two vectors broadly point the same way, zero when they are perpendicular, negative when they oppose. Example: (3, 4)·(2, −1) = 6 − 4 = 2, a small positive number, meaning the two arrows lean the same way but only just. Change b to (4, −3) and the product is 12 − 12 = 0: exactly perpendicular, which is far easier to test this way than by measuring any angle.

The notation is younger than the idea. Through the 1880s Hamilton's quaternions were the standard tool for three-dimensional algebra, and their product mixed a scalar part with a vector part in one unwieldy object. Josiah Willard Gibbs at Yale, printing lecture notes privately for his students, and Oliver Heaviside in England, working on Maxwell's equations, independently tore that product in half: keeping the scalar piece as the dot product and the vector piece as the cross product. Quaternion loyalists fought the change bitterly in the pages of Nature; Gibbs and Heaviside won, and every physics course since teaches their version.