vectordifference

Vectordifference is the operation of subtracting one vector from another. Given two vectors v and w in the same vector space, their difference is v - w, obtained by subtracting corresponding components. In coordinate form, if v = (v1, v2, ..., vn) and w = (w1, w2, ..., wn), then v - w = (v1 - w1, v2 - w2, ..., vn - wn).

Geometrically, v - w represents the vector from the point represented by w to the point represented by v. It can also be written as v + (-w), since subtraction is addition with the additive inverse of w. The operation is not commutative; in general v - w ≠ w - v. It is, however, associative with addition in the sense that (v - w) + w = v and v - (w + u) = (v - w) - u, reflecting its basis in the additive structure of a vector space. The norm of the difference, ||v - w||, gives the distance between the two vectors when they are considered as points in Euclidean space.

In computation, vectordifference is performed element-wise. It requires that v and w have the same dimension. Applications include determining relative motion in physics (relative velocity), computing residuals in data analysis, and measuring similarity or separation between data points in machine learning and computer graphics.

Common notations include v - w, Δv in context, or simply the subtraction operation in a vector space.