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I.1 Introduction I.1.1 Vector Algebra Vector Terminology Scalars are mathematics quantities that can be fully defined by specifying their magnitude in suitable units of measure. The mass is a scalar and can be expressed in kilograms, the time is a scalar and can be expressed seconds, and the temperature can be expressed in degrees. 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 number. 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. To distinguish vectors from scalars it is customary to denote vectors by boldface letters. Thus, the vector shown in Figure I.1.1(a) is denoted by r or r AB . The symbol | r | = r represents the magnitude (or module, or absolute value) of the vector r . In handwritten work a distinguishing mark is used for vectors, such as an arrow over the symbol, r or −→ AB , a line over the symbol, ¯ r , or an underline, r . The vectors are depicted by either straight or curved arrows. A vector represented by a straight arrow has the direction indicated by the arrow. The direction of a vector represented by a curved arrow 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. 3
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a b a b r A (a) (b) B FIGURE I.1.1 Vector representations: (a) straight arrow and (b) straight and curved arrows. Figure I.1.1(b) shows representations of vectors. Sometimes vectors are represented by means of a straight or curved arrow together with a measure number. In this case the vector is regarded as having the direction indicated by the arrow if the measure number is positive, and the opposite direction if it is negative. A bound (or fixed ) vector is a vector associated with a particular point P in space (Fig. I.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 application can be represented as the tail [Fig. I.1.2(a)] or the head of the vector arrow [Fig. I.1.2(b)]. A free vector is not associated with a particular point or line in space. A transmissible (or sliding ) vector is a vector that can be moved along its line of action without change of meaning.
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