vector

A vector is the mathematical object used to describe anything that has both a size and a direction: velocity, force, acceleration, displacement, electric and magnetic fields. Drawn as an arrow, its length stands for the magnitude and its orientation stands for the direction. In two or three dimensions, a vector is conveniently written as an ordered list of components — its projections onto the chosen axes. The magnitude is recovered from those components by Pythagoras: |v| = √(vₓ² + v_y² + v_z²).

Two operations dominate. The first is addition: to combine two vectors, place them tip-to-tail — the resultant runs from the starting point of the first to the ending point of the second. Equivalently, add the components. Geometrically, if you draw a parallelogram whose sides are the two vectors, the resultant is the diagonal. The second is scalar multiplication: stretching or compressing a vector by a number, with a negative scalar flipping the direction.

Vectors are the language in which classical mechanics is written once motion leaves the one-dimensional line. A projectile's velocity has a horizontal component that stays constant and a vertical component that changes with gravity; the total velocity is the vector sum of the two, and the speed is its magnitude. In higher-dimensional physics — relativity, electromagnetism, quantum mechanics — vectors generalise to tensors and state-vectors, but the elementary two- and three-dimensional arrows are still the working tools of everyday problems.

The idea of a directed quantity existed informally in Newton's Principia as the 'parallelogram of forces'. The modern notation is much younger: William Rowan Hamilton introduced quaternions in 1843 as a way to multiply three-dimensional directed quantities, and Josiah Willard Gibbs and Oliver Heaviside in the 1880s stripped the quaternion formalism down to what we now call vector algebra — dot product, cross product, and component notation.