Vector Addition is Associative/Proof 1

Theorem

Let $\mathbf a, \mathbf b, \mathbf c$ be vectors.

Then:

$\mathbf a + \paren {\mathbf b + \mathbf c} = \paren {\mathbf a + \mathbf b} + \mathbf c$

where $+$ denotes vector addition.

Proof

Let $\mathbf a$, $\mathbf b$ and $\mathbf c$ be positioned in space so they are end to end as in the above diagram.

Let $\mathbf v$ be a vector representing the closing side of the polygon whose other $3$ sides are represented by $\mathbf a$, $\mathbf b$ and $\mathbf c$.

By the Parallelogram Law we can add any pair of vectors, and add a third vector to their resultant.

Hence we have:

\(\ds \mathbf v\)

\(=\)

\(\ds \mathbf a + \mathbf b + \mathbf c\)

\(\ds \)

\(=\)

\(\ds \paren {\mathbf a + \mathbf b} + \mathbf c\)

\(\ds \)

\(=\)

\(\ds \mathbf a + \paren {\mathbf b + \mathbf c}\)

\(\ds \)

\(=\)

\(\ds \paren {\mathbf a + \mathbf c} + \mathbf b\)

$\blacksquare$

Sources

• 1951: B. Hague: An Introduction to Vector Analysis (5th ed.) ... (previous) ... (next): Chapter $\text I$: Definitions. Elements of Vector Algebra: $3$. Addition and Subtraction of Vectors