# Set of Vectors defined by Directed Line Segments in Space forms Vector Space

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## Theorem

Let $\R^3$ be a real cartesian space of $3$ dimensions.

Consider the set $S$ of directed line segments in $\R^3$.

Let the equivalence relation $\sim$ be applied to $\R^3$ such that:

- $\forall L_1, L_2 \in \R^3: L_1 \sim L_2$ if and only if there exists a translation $T$ such that $T \left({L_1}\right) = L_2$

Let $\mathbb V$ denote the set of equivalence classes of $\sim$ on $S$.

The elements of $\mathbb V$ form a vector space where:

- $\forall \mathbf v_1, \mathbf v_2 \in \mathbb V: \mathbf v_1 + \mathbf v_2$ denotes the element of $\mathbb V$ exemplified by the result of the operation of joining of a representative element of element of $\mathbf v_2$ to the end of a similarly representative element of element of $\mathbf v_1$.
- $\forall \mathbf v \in \mathbb V, \lambda \in \R: \lambda \mathbf v$ denotes the element of $\mathbb V$ exemplified by the result of the operation of mulitplying a representative element of element of $\mathbf v$ by the scale factor $\lambda$.

## Proof

## Sources

- 1961: I.M. Gel'fand:
*Lectures on Linear Algebra*(2nd ed.) ... (previous) ... (next): $\S 1$: $n$-Dimensional vector spaces: Definition $1$