Definition:Engineering Notation

Euclidean 2-space
Define the ordered 2-tuples:


 * $\mathbf{i} = \langle{1,0}\rangle$


 * $\mathbf{j} = \langle{0,1}\rangle$

From Standard Ordered Basis, we have that any vector in $\R^2$ can be represented by:


 * $c_1\mathbf{i} + c_2\mathbf{j}$.

where $c_1,c_2 \in \R$.

This way of presenting vectors is called Engineering Notation.

Euclidean 3-space
Define the ordered 3-tuples:


 * $\mathbf{i} = \langle{1,0,0}\rangle$


 * $\mathbf{j} = \langle{0,1,0}\rangle$


 * $\mathbf{k} = \langle{0,0,1}\rangle$

By the same logic as the above definition, we can write any vector in $\R^3$ as:


 * $c_1\mathbf{i} + c_2\mathbf{j} + c_3\mathbf{k}$.

where $c_1,c_2,c_3 \in \R$.

Note that $\mathbf{i}$ and $\mathbf{j}$ take on a different meaning in 3-space than in 2-space.

Euclidean n-space
In higher dimensions, rather than writing $\mathbf{l},\mathbf{m},\mathbf{n}$, and so on, the convention is to use:


 * $e_1 = \langle{1,0,0,\cdots,0}\rangle$


 * $e_2 = \langle{0,1,0,\cdots,0}\rangle$


 * $\vdots$


 * $e_n = \langle{0,0,0,\cdots,1}\rangle$

And then any vector in $\R^n$ can be expressed as:


 * $c_1e_1 + c_2e_2 + \cdots + c_ne_n$.

where $c_1,c_2,\cdots,c_n \in \R$.