Vector Addition is Associative/Proof 2

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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

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

$\blacksquare$


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