Vector Addition is Associative

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 1

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 $

$=$

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

$\blacksquare$

Proof 2

Let $\mathbf a$, $\mathbf b$ and $\mathbf c$ be expressed in component form:

$\ds \mathbf a$

$=$

$\ds a_1 \mathbf e_1 + a_2 \mathbf e_2 + \dotsb + a_n \mathbf e_n$

$\ds \mathbf b$

$=$

$\ds b_1 \mathbf e_1 + b_2 \mathbf e_2 + \dotsb + b_n \mathbf e_n$

$\ds \mathbf c$

$=$

$\ds c_1 \mathbf e_1 + c_2 \mathbf e_2 + \dotsb + c_n \mathbf e_n$

$\ds \leadsto \ \ $

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

$=$

$\ds \paren {\sum_{j \mathop = 1}^n a_j \mathbf e_j + \sum_{j \mathop = 1}^n b_j \mathbf e_j} + \sum_{j \mathop = 1}^n c_j \mathbf e_j$

$\ds $

$=$

$\ds \sum_{j \mathop = 1}^n \paren {a_j + b_j} \mathbf e_j + \sum_{j \mathop = 1}^n c_j \mathbf e_j$

Scalar Multiplication of Vectors is Distributive over Vector Addition

$\ds $

$=$

$\ds \sum_{j \mathop = 1}^n \paren {\paren {a_j + b_j} + c_j} \mathbf e_j$

Scalar Multiplication of Vectors is Distributive over Vector Addition

$\ds $

$=$

$\ds \sum_{j \mathop = 1}^n \paren {a_j + \paren {b_j + c_j} } \mathbf e_j$

Associative Law of Addition

$\ds $

$=$

$\ds \sum_{j \mathop = 1}^n a_j \mathbf e_j + \sum_{j \mathop = 1}^n \paren {b_j + c_j} \mathbf e_j$

Scalar Multiplication of Vectors is Distributive over Vector Addition

$\ds $

$=$

$\ds \sum_{j \mathop = 1}^n a_j \mathbf e_j + \paren {\sum_{j \mathop = 1}^n b_j + \sum_{j \mathop = 1}^n c_j \mathbf e_j}$

Scalar Multiplication of Vectors is Distributive over Vector Addition

$\ds $

$=$

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

Scalar Multiplication of Vectors is Distributive over Vector Addition

$\blacksquare$

Sources

1921: C.E. Weatherburn: Elementary Vector Analysis ... (previous) ... (next): Chapter $\text I$. Addition and Subtraction of Vectors. Centroids: Addition and subtraction of vectors: $4$. Component and Resultant
1927: C.E. Weatherburn: Differential Geometry of Three Dimensions: Volume $\text { I }$ ... (previous) ... (next): Introduction: Vector Notation and Formulae
1957: D.E. Rutherford: Vector Methods (9th ed.) ... (previous) ... (next): Chapter $\text I$: Vector Algebra: $\S 1$.
1968: Murray R. Spiegel: Mathematical Handbook of Formulas and Tables ... (previous) ... (next): $\S 22$: Laws of Vector Algebra: $22.2$
1998: David Nelson: The Penguin Dictionary of Mathematics (2nd ed.) ... (previous) ... (next): vector
2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): vector
2014: Christopher Clapham and James Nicholson: The Concise Oxford Dictionary of Mathematics (5th ed.) ... (previous) ... (next): addition (of vectors)
2021: Richard Earl and James Nicholson: The Concise Oxford Dictionary of Mathematics (6th ed.) ... (previous) ... (next): addition (of vectors)