Definition:Engineering Notation

Euclidean 2-space
Define the ordered 2-tuples:


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


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

From Standard Ordered Basis is 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:


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


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


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

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:


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


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


 * $\vdots$


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

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


 * $c_1 \mathbf e_1 + c_2 \mathbf e_2 + \cdots + c_n \mathbf e_n$

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

This convention is also frequently seen for $2$-space and $3$-space.

Comment
Particularly in hand-written material, it is common to put a circumflex above the letters, as is common to do with other unit vectors.

With such a notation, they may be referred to as i-hat, j-hat, and k-hat.

The "hat" can be used as a diacritic in addition to or instead of the dots above the letters i and j.