c1_2110 - Chapter 1 Vector Algebra 1.1 Terminology and...

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Chapter 1 Vector Algebra 1.1 Terminology and Notation Scalars are mathematics quantities that can be fully defined by specifying their mag- nitude in suitable units of measure. Mass is a scalar quantity and can be expressed in kilograms, time is a scalar and can be expressed in seconds, and temperature is a scalar quantity that can be expressed in degrees Celsius. Vectors are quantities that require the specification of magnitude, orientation, and sense. The characteristics of a vector are the magnitude, the orientation, and the sense. The magnitude of a vector is specified by a positive number and a unit having appropriate dimensions. No unit is stated if the dimensions are those of a pure num- ber. The orientation of a vector is specified by the relationship between the vector and given reference lines and/or planes. The sense of a vector is specified by the order of two points on a line parallel to the vector. Orientation and sense together determine the direction of a vector. The line of action of a vector is a hypothetical infinite straight line collinear with the vector. Displacement, velocity, and force are examples of vectors quantities. To distinguish vectors from scalars it is customary to denote vectors by boldface letters Thus, the displacement vector from point A to point B could be denoted as r or r AB . The symbol | r | = r represents the magnitude (or module, norm, or absolute value) of the vector r . In handwritten work a distinguishing mark is used for vec- tors, such as an arrow over the symbol, -→ r or AB , a line over the symbol, ¯ r , or an underline, r . The vectors are most frequently depicted by straight arrows. A vector represented by a straight arrow has the direction indicated by the arrow. The displacement vector from point A to point B is depicted in Fig. 1.1(a) as a straight arrow. In some cases it is necessary to depict a vector whose direction is perpendicular to the surface 1
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2 1 Vector Algebra in which the representation will be drawn. Under this circumstance the use of a portion of a circle with a direction arrow is useful. The orientation of the vector is perpendicular to the plane containing the circle and the sense of the vector is the same as the direction in which a right-handed screw moves when the axis of the screw is normal to the plane in which the arrow is drawn and the screw is rotated as indicated by the arrow. Figure 1.1(b) uses this representation to depict a vector directed out of the reading surface toward the reader. v r A (a) (b) B Fig. 1.1 Representations of vectors A bound vector is a vector associated with a particular point P in space (Fig. 1.2). The point P is the point of application of the vector, and the line passing through P and parallel to the vector is the line of action of the vector. The point of appli- cation may be represented as the tail, Fig. 1.2(a), or the head of the vector arrow, Fig. 1.2(b). A free vector is not associated with any particular point in space. A transmissible (or sliding ) vector is a vector that can be moved along its line of ac- tion without change of meaning.
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This note was uploaded on 08/29/2011 for the course MECH 2110 taught by Professor Clark,b during the Spring '08 term at Auburn University.

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c1_2110 - Chapter 1 Vector Algebra 1.1 Terminology and...

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