Vector notation

From Wikipedia, the free encyclopedia

For information on vectors as a mathematical object see vector (spatial). This page is about notation of vectors.

[edit] Declaration

A vector can be declared in three ways:

Parentheses can enclose an ordered set of coordinates: $(1,2,3)$ .
Angle braces can also enclose an ordered set: $\big\langle 1,2,3 \big\rangle$
Unit vectors can be used to describe a vector more algebraically: $1 \mathbf{i} + 2 \mathbf{j} + 3 \mathbf{k}$ where $\mathbf{i}, \mathbf{j}, \mbox{and} \; \mathbf{k}$ are the unit vectors in each of the three dimensions.

A threespace vector was used for these examples, but the first two methods can be applied to any vector space. The unit vector notation however, is only common for 2 and 3 dimensional vectors as there are not standard unit vectors for higher dimensioned spaces.

Letters representing vector quantities are distinguished from scalar quantities by bolding them, for example ω represents the magnitude of a rotational velocity while ω represents a rotational velocity. When handwritten this is difficult to achieve, so several different notations are used. These include writing a tilde over or under the letter, and writing an arrow over the letter.

The origins of this come from the typographical convention of tilde-shaped or wavy underlining to represent bolding of text. Straight underlines are often lazily used to represent vectors but in typography these represent the italicising of charaters.

[edit] Products

There are three vector multiplications:

The cross product is notated with the multiplication cross: $\mathbf{a} \times \mathbf{b}$
The dot product is notated with the multiplication dot: $\mathbf{a} \cdot \mathbf{b}$
Scalar multiplication is usually written implicitly to avoid confusion with the other two types of multiplication: $(c)\mathbf{a}$