In general, in one dimension—as well as in higher dimensions, such as in a plane or in space—we can add any number of vectors and we can do so in any order because the addition of vectors is commutative,

We used the distributive property in Equation 2.4 and Equation 2.6.

When adding many vectors in one dimension, it is convenient to use the concept of a unit vector. A unit vector, which is denoted by a letter symbol with a hat, such as \(\hat{u}\), has a magnitude of one and does not have any physical unit so that |\(\hat{u}\)| ≡ u = 1. The only role of a unit vector is to specify direction. For example, instead of saying vector \(\vec{D}_{AB}\) has a magnitude of 6.0 km and a direction of northeast, we can introduce a unit vector \(\hat{u}\) that points to the northeast and say succinctly that \(\vec{D}_{AB}\) = (6.0 km) \(\hat{u}\). Then the southwesterly direction is simply given by the unit vector\(- \hat{u}\). In this way, the displacement of 6.0 km in the southwesterly direction is expressed by the vector

$$\vec{D}_{BA} = (−6.0\; km)\; \hat{u} \ldotp$$

Contributors

Samuel J. Ling (Truman State University), Jeff Sanny (Loyola Marymount University), and Bill Moebs with many contributing authors. This work is licensed by OpenStax University Physics under a Creative Commons Attribution License (by 4.0).