Vector notation

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

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.

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: $(c)\mathbf {a}$