# 2.2: Scalars and Vectors (Part 1)

- Page ID
- 3971

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**,

$$\vec{A} + \vec{B} = \vec{B} + \vec{A} \ldotp \tag{2.7}$$

and **associative,**

$$ (\vec{A} + \vec{B}) + \vec{C} = \vec{A} + (\vec{B} + \vec{C}) \ldotp \tag{2.8}$$

Moreover, multiplication by a scalar is **distributive**:

$$ \alpha_{1} \vec{A} + \alpha_{2} \vec{A} = (\alpha_{1} + \alpha_{2}) \vec{A} \ldotp \tag{2.9}$$

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).