Scalars & vectors

A scalar is a quantity with magnitude only; a vector is a quantity with both magnitude and direction. In latin, the word vector is a noun with the literal meaning carrier—this is a good way to think about them in math. A vector is an arrow that takes you from one place to another (the specific places are not part of the vector, though). For example, a speed of 50 km/h is a scalar while a velocity of 50 km/h due north is a vector.

Here are some examples of scalar and vector quantities:

We represent scalars using ordinary variable notation like $a\text{,}$ $x\text{,}$ $v\text{,}$ or $t\text{.}$ Vectors use a different notation: either boldface, like $𝐯\text{,}$ or arrows, like $\overrightarrow{v}\text{.}$ The arrow notation is always used in handwriting. If you want to designate a vector from point A to point B, use $\overrightarrow{AB}\text{.}$

We use absolute value notation when we want to get the magnitude of a vector. For example, if $\overrightarrow{v}$ is 5 km/h [N25ºE], then $\left|\overrightarrow{v}\right|$ is 5 km/h.

Two vectors are equal if and only if they have the same magnitude and the same direction. In other words, they must be parallel and have the same length. Opposite vectors have the same magnitude but opposite directions. For example, $\overrightarrow{AB}$ and $\overrightarrow{BA}$ are opposite due to the identity $\overrightarrow{AB}=-\overrightarrow{BA}\text{.}$