Chapter 1Vector Algebra1.1 Terminology and NotationScalars 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 expressedin kilograms, time is a scalar and can be expressed in seconds, and temperature is ascalar 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 thesense.The magnitude of a vector is specified by a positive number and a unit havingappropriate 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 vectorand given reference lines and/or planes.The sense of a vector is specified by the order of two points on a line parallel tothe 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 withthe vector.Displacement, velocity, and force are examples of vectors quantities.To distinguish vectors from scalars it is customary to denote vectors by boldfaceletters Thus, the displacement vector from point A to point B could be denoted as ror rAB. The symbol |r| = r represents the magnitude (or module, norm, or absolutevalue) 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 anunderline, r.The vectors are most frequently depicted by straight arrows. A vector representedby a straight arrow has the direction indicated by the arrow. The displacement vectorfrom point A to point B is depicted in Fig. 1.1(a) as a straight arrow. In some casesit is necessary to depict a vector whose direction is perpendicular to the surface12 1 Vector Algebrain which the representation will be drawn. Under this circumstance the use of aportion of a circle with a direction arrow is useful. The orientation of the vector isperpendicular to the plane containing the circle and the sense of the vector is thesame as the direction in which a right-handed screw moves when the axis of thescrew is normal to the plane in which the arrow is drawn and the screw is rotatedas indicated by the arrow. Figure 1.1(b) uses this representation to depict a vectordirected out of the reading surface toward the reader.vrA(a)(b)BFig. 1.1 Representations of vectorsA 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 throughP 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. Atransmissible (or sliding) vector is a vector that can be moved along its line of ac-tion without change of meaning.vPvP(b)(a)line of actionbound vectorline of actionbound vectorpoint of applicationpoint of applicationFig. 1.2 Bound or fixed vector: (a) point of application represented as the tail of the vector arrowand (b) point of application represented as the head of the vector arrowTo move the rigid body in Fig. 1.3 the force vector F can be applied anywherealong the line ∆ or may be applied at specific points A, B and C. The force vector Fis a transmissible vector because the resulting motion is the same in all cases.1.1 Terminology and Notation 3F ABCF F Δ Δ body Fig. 1.3 Transmissible vector: the force vector F can be applied anywhere along the line ∆If the body is not rigid, the force F applied at A will cause a different deforma-tion of the body than F applied at a different point B. If one is interested in thedeformation of the body, the force F positioned at C is a bound vector.The operations of vector analysis deal only with the characteristics of vectors andapply, therefore, to bound, free, and transmissible vectors.EqualityTwo vectors a and b are said to be equal to each other when they have the samecharacteristics. One then writesa = b. (1.1)Equality does not imply physical equivalence. For instance, two forces representedby equal vectors do not necessarily cause identical motions of a body on which theyact.Product of a Vector and a ScalarThe product of a vector v and a scalar s, sv or vs, is a vector having the followingcharacteristics:1. Magnitude. |sv|≡|vs|= |s||v|, where |s|= s denotes the absolute value (or mag-nitude, or module) of the scalar s.2. Orientation. sv is parallel to v. If s = 0, no definite orientation is attributed to sv.3. Sense. If s > 0, the sense of sv is the same as that of v. If s < 0, the sense of sv isopposite to that of v. If s = 0, no definite sense is attributed to sv.Zero VectorA zero vector is a vector that does not have a definite direction and whose magnitudeis equal to zero. The symbol used to denote a zero vector is 0.Unit VectorA unit vector is a vector with magnitude equal to 1. Given a vector v, a unit vectoru having the same direction as v is obtained by forming the product of v with thereciprocal of the magnitude of vu = v1|v|=v|v|. (1.2)4 1 Vector AlgebraVector AdditionThe sum of a vector v1and a vector v2: v1+ v2or v2+ v1is a vector whose charac-teristics can be found by either graphical or analytical processes. The vectors v1andv2add according to the parallelogram law: the vector v1+ v2is represented by thediagonal of a parallelogram formed by the graphical representation of the vectors,see Fig. 1.4(a).v1v2v1v2||||(a)+v2v1v1v2(b)-v2v1-v2v1-v2v1-v2v1v2+v2v1(c)(d)Fig. 1.4 Vector addition: (a) parallelogram law, (b) moving the vectors successively to parallelpositions. Vector difference: (c) parallelogram law, (d) moving the vectors successively to parallelpositionsThe vector v1+ v2is called the resultant of v1and v2. The vectors can be addedby moving them successively to parallel positions so that the head of one vectorconnects to the tail of the next vector. The resultant is the vector whose tail connectsto the tail of the first vector, and whose head connects to the head of the last vector,see Fig. 1.4(b).The sum v1+(−v2) is called the difference of v1and v2and is denoted by v1−v2,see Figs. 1.4(c) and 1.4(d). The sum of n vectors vi, i = 1,. ..,n,n∑i=1vior v1+ v2+ .. . + vnis called the resultant of the vectors vi, i = 1,.