Definition:Engineering Notation

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Euclidean 2-space

Define the ordered 2-tuples:

\(\ds \mathbf i\) \(=\) \(\ds \tuple {1, 0}\)
\(\ds \mathbf j\) \(=\) \(\ds \tuple {0, 1}\)

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:

\(\ds \mathbf i\) \(=\) \(\ds \tuple {1, 0, 0}\)
\(\ds \mathbf j\) \(=\) \(\ds \tuple {0, 1, 0}\)
\(\ds \mathbf k\) \(=\) \(\ds \tuple {0, 0, 1}\)

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:

\(\ds \mathbf e_1\) \(=\) \(\ds \tuple {1, 0, 0, \ldots, 0, 0}\)
\(\ds \mathbf e_2\) \(=\) \(\ds \tuple {0, 1, 0, \ldots, 0, 0}\)
\(\ds \) \(\vdots\) \(\ds \)
\(\ds \mathbf e_n\) \(=\) \(\ds \tuple {0, 0, 0, \ldots, 0, 1}\)

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.

Also denoted as

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

$\hat \imath, \hat \jmath, \hat k$

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 $j$.