Vector Addition is Associative
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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: $22.2$
- 2014: Christopher Clapham and James Nicholson: The Concise Oxford Dictionary of Mathematics (5th ed.) ... (previous) ... (next): addition (of vectors)