Addition (Vector)

linear-algebra

Definition

Addition (Vector)

The sum of two vectors $\mathbf{a},\mathbf{b}\in V$, where $V$ is a vector space over a field $F$, is the vector obtained by adding components entrywise:

$$\mathbf{a}+\mathbf{b}=\left[\begin{array}{c}{\mathbf{a}}_1\\ {\mathbf{a}}_2\\ ⋮\\ {\mathbf{a}}_n\end{array}\right]+\left[\begin{array}{c}{\mathbf{b}}_1\\ {\mathbf{b}}_2\\ ⋮\\ {\mathbf{b}}_n\end{array}\right]=\left[\begin{array}{c}{\mathbf{a}}_1+{\mathbf{b}}_1\\ {\mathbf{a}}_2+{\mathbf{b}}_2\\ ⋮\\ {\mathbf{a}}_n+{\mathbf{b}}_n\end{array}\right]$$

Geometric Interpretation

Geometrically, vector addition is the parallelogram law: place the tails of $\mathbf{a}$ and $\mathbf{b}$ together; their sum is the diagonal of the parallelogram they span.