Definition

If 2 vectors are in the same \(n\)-dimensional space, then they can be added. Addition happens cooridnate-wise. In \(n\)-dimensional space, the formula is the following: \[< a_1, a_2, \dots, a_n > + < b_1, b_2, \dots, b_n > = < a_1 + b_1, a_2 + b_2, \dots, a_n + b_n >\]

For an example in 2-dimensions: \begin{align} <2,3>+<4,6> & = <2+4, 3+6> \\ & = <6, 9> \end{align} There is no definition of a sum of 2 vectors of different dimensions.
\[<3,2> + <1,3,4> = \text{undefined}\]

Properties of the Vector Addition

• Commutativity: If \(\overrightarrow{x}\) and \(\overrightarrow{y}\) are \(n\)-dimensional vectors, then \[\overrightarrow{x}+\overrightarrow{y}=\overrightarrow{y}+\overrightarrow{x}.\]
• Identity: There is a zero: The vector \((0,0,0,\dots,0)\) with \(n\) 0's is the \(n\)-dimensional \(0\) vector. It is written \(\overrightarrow{0}.\) For every \(n\)-dimensional vector \(\overrightarrow{x},\) \[\overrightarrow{x}+\overrightarrow{0}=\overrightarrow{x}.\]
• Associativity: For any three \(n\)-dimensional vectors \(\overrightarrow{x},\) \(\overrightarrow{y},\) and \(\overrightarrow{z},\) \[(\overrightarrow{x}+\overrightarrow{y})+\overrightarrow{z} = \overrightarrow{x}+(\overrightarrow{y}+\overrightarrow{z})\]