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3.2: Scalars and Vectors (Part 1)

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  • A photo of a dog. Below the photo is a horizontal arrow which starts below the dog’s tail and ends below the dog’s nose. The arrow is labeled Vector D, and its length is labeled as magnitude D. The start (tail) of the arrow is labeled “From rail of a vector origin” and its end (head) is labeled “To head of a vector end.”
    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

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


    • 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).