Basics of Vectors: Scalars, Vectors, and Unit Vector

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Scalar quantities have magnitude only, while vector quantities have both magnitude and direction. Unit vectors have magnitude of 1 and direction. Vector addition and subtraction involve combining vectors component by component. Various rules and laws govern vector algebra.

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Basics of Vectors Part I

SCALARS AND VECTORS: There are two classes of physical quantities, each with its characteristic properties and an appropriate algebra. Scalar quantities: These have magnitude only and do not involve direction. To specify a scalar quantity completely, it is necessary to know a) a unit of the same kind and b) a number stating how many times the unit is contained in the quantity, i.e. a scalar can be specified by a single number. Example: mass, time, volume, density, temperature, electric potential, charge They are indicated by non-bold face letters, for instance, S .

SCALARS AND VECTORS: Vector quantities: These have both magnitude and direction. To specify a vector quantity completely, it is necessary to know a) a unit of the same kind and b) a number giving the magnitude of quantity in terms of this unit. c) a statement of the direction. Example: velocity, force, displacement, and electric field intensity They are indicated by bold face letters, for instance, V or by a letter with an arrow on top of it, such as ?. Another class of physical quantities is called tensors, of which scalars and vectors are special cases.

UNIT VECTOR: A unit vector ??along ? is defined as a vector whose magnitude is unity (i.e., 1) and its direction is along ?, that is,. ? ? ? ? ??= = Note that ??=1. Thus we may write ? as: ? = ? ?? A vector ? in cartesian (or rectangular) coordinate may be represented as : (??,??,??) ?? ??+ ?? ??+ ?? ?? or ??2+ ??2+ ??2 Magnitude of vector ? is given by: ? =

VECTOR ADDITION AND SUBTRACTION: Two vectors A and B can be added together to give another vector C; that is, C = A + B The vector addition is carried out component by component. Thus, if ? = (??,??,??) and ? = (??,??,??). ? = ??+ ????+ ??+ ????+ ??+ ???? Vector subtraction is similarly carried out as ? = ? ? = ? + ( ?) ? = ?? ????+ ?? ????+ ?? ????

Graphically, vector addition and subtraction are obtained by either the parallelogram rule or the head-to-tail rule as portrayed in Figures 1.2 and 1.3, respectively. FIGURE 1.2 Vector addition C = A + B: (a) parallelogram rule, (b) head-to-tail rule.

FIGURE 1.3 Vector subtraction D = A - B: (a) parallelogram rule, (b) head-to-tail rule.

The three basic laws of algebra obeyed by any given by vectors A, B, and C, are summarized as follows: Law Addition Commutative A + B = B + A Associative A + (B + C) = (A + B) + C Distributive k(A+ B) = kA+ kB where k is scalar.

POSITION VECTOR The position vector??(or radius vector) of point P is as the directed distance from the origin O to P: i.e.. ??= ?? = ???+ ???+ ??? FIGURE 1.4 Illustration of position vector rP = 3ax + 4ay + 5az.

DISTANCE VECTOR The distance vector is the displacement from one point to another. If two points P and Q are given by (??,??,??) and (??,??,??), the distance vector (or separation vector) is the displacement from P to Q as shown in Figure 1.5; that is, ??+ ???= ?? ???= ?? ?? ???= (?? ??)??+ (?? ??) ??+ (?? ??) ?? FIGURE 1.5 Distance vector rPQ.