Vector Addition¶

Definition¶

Vector addition is the operation $+\colon V\times V\to V$ in a vector space that combines two vectors to produce another vector.

Interpretation¶

Vector addition combines two vectors of the same space into a third one. In coordinate spaces it is done coordinate by coordinate, $(u_1,\ldots,u_n)+(v_1,\ldots,v_n)=(u_1+v_1,\ldots,u_n+v_n)$ $v_{n})$ , so it only makes sense for vectors with the same number of coordinates.

The vectors u = (3, 1) and v = (1, 2) added tip to tail, with u + v = (4, 3) as the diagonal of the parallelogram they span.

In the figure, $\boldsymbol{u}=(3,1)$ and $\boldsymbol{v}=(1,2)$ add to $(3+1,1+2)=(4,3)$ $(4, 3)$ . Geometrically, $\boldsymbol{v}$ is drawn again at the tip of $\boldsymbol{u}$ , and $\boldsymbol{u}+\boldsymbol{v}$ runs from the origin to the end of this path: the tip-to-tail rule. It is also the diagonal of the parallelogram with sides $\boldsymbol{u}$ and $\boldsymbol{v}$ .

Drag $\boldsymbol{u}$ and $\boldsymbol{v}$ . Adding coordinate by coordinate is the same as placing $\boldsymbol{v}$ at the tip of $\boldsymbol{u}$ ; the other route, $\boldsymbol{u}$ first at the tip of $\boldsymbol{v}$ , ends at the same point.

Going round the parallelogram the other way, $\boldsymbol{v}$ first, ends at the same point: this is commutativity, $\boldsymbol{u}+\boldsymbol{v}=\boldsymbol{v}+\boldsymbol{u}$ $u$ . With associativity, the zero vector and the inverses $-\boldsymbol{v}$ , addition makes $(V,+)$ an Abelian group. In other vector spaces addition is whatever the space defines; for polynomials, coefficients of equal powers are added.

Key Results¶

Vector addition is associative and commutative.
The zero vector is the additive identity.

Examples¶

In $\mathbb{R}^2$ , $(2,3)+(-1,5)=(1,8)$ $(1, 8)$ .

Prerequisites¶

Vector Space

Sources¶

Math II: Linear Algebra · Week 2

Aliases: addition of vectors Symbols: real numbers

Last updated: 20 October 2026 03:02