We often use mathematics to describe physical situations. Two types of quantities that are particularly important for describing physical systems are scalars and vectors. In the notes below, you will read about those quantities (in general) and their properties.

Scalars are quantities that can be represented by a single number. Typical examples include mass, volume, density, and speed.

Vectors are quantities that have both a magnitude and direction. Typical examples include displacement, velocity, momentum, and force.

Vectors are often represented with arrows. The end with the triangle is the “tip” or “head.” The other end is called the “tail.” The tail of a vector can be located anywhere; it is the difference between the tip and the tail that defines the vector itself. To the right is an example of a typical representation (a diagram) of a vector with the tip and tail labeled. We have no such diagrammatic representations for scalars.

We define vectors in three dimensional space relative to some origin (where the tail of the vector is located). For example, a position vector $\vec{r}$ might defined relative to the origin of coordinates. The measures of the vector along the coordinate axes are called the vector's “components,” which can be positive or negative. Mathematically, a vector can be written with “bracket” notation:

$$ \mathbf{r} = \vec{r} = \langle r_x, r_y, r_z \rangle $$

where $r_x$, $r_y$, and $r_z$ are the vector components in the $x$, $y$, and $z$ direction respectively. They tell you “how much” of the vector $\vec{r}$ is aligned with each coordinate direction. The vector itself