Definition:Engineering Notation
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Definition
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 $j$.
Sources
- For a video presentation of the contents of this page, visit the Khan Academy.