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 |
$\blacksquare$
Sources
- 1951: B. Hague: An Introduction to Vector Analysis (5th ed.) ... (previous) ... (next): Chapter $\text I$: Definitions. Elements of Vector Algebra: $4$. Components of a Vector