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# 3.2: Scalars and Vectors (Part 1)

• • Contributed by OpenStax
• General Physics at OpenStax CNX Figure 2.7(c)) or tail-to-tail fashion. The magnitude of the vector difference, then, is the absolute value D = |A − B| of the difference of their magnitudes. The direction of the difference vector $$\vec{D}$$ is parallel to the direction of the longer vector.

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).