Subtraction (Vector)

linear-algebra

Definition

Subtraction (Vector)

The difference of two vectors $\mathbf{a},\mathbf{b}\in V$, where $V$ is a vector space over a field $F$, is the vector obtained by subtracting 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 subtraction is the reverse diagonal of the parallelogram: place the tails of $\mathbf{a}$ and $\mathbf{b}$ together; their difference $\mathbf{a}-\mathbf{b}$ is the vector from the tip of $\mathbf{b}$ to the tip of $\mathbf{a}$.