VECTOR ALGEBRA

1 MATHEMATICS424 In most sciences one generation tears down what another has built and whatone has established another undoes. In Mathematics alone each generationbuilds a new story to the old structure. HERMAN HANKEL IntroductionIn our day to day life, we come across many queries suchas What is your height? How should a football player hitthe ball to give a pass to another player of his team? Observethat a possible answer to the first query may be meters,a quantity that involves only one value (magnitude) whichis a real number.

2 Such quantities are called , an answer to the second query is a quantity (calledforce) which involves muscular strength (magnitude) anddirection (in which another player is positioned). Suchquantities are called vectors. In mathematics, physics andengineering, we frequently come across with both types ofquantities, namely, scalar quantities such as length, mass,time, distance, speed, area, volume, temperature, work,money, voltage, density, resistance etc. and VECTOR quantities like displacement, velocity,acceleration, force, weight, momentum, electric field intensity this chapter, we will study some of the basic concepts about vectors, variousoperations on vectors, and their algebraic and geometric properties.

3 These two type ofproperties, when considered together give a full realisation to the concept of vectors,and lead to their vital applicability in various areas as mentioned Some Basic ConceptsLet l be any straight line in plane or three dimensional space. This line can be giventwo directions by means of arrowheads. A line with one of these directions prescribedis called a directed line (Fig (i), (ii)).Chapter10 VECTOR Hamilton(1805-1865) NCERTnot to be republishedVECTOR ALGEBRA425 Now observe that if we restrict the line l to the line segment AB, then a magnitudeis prescribed on the line l with one of the two directions, so that we obtain a directedline segment (Fig (iii)).

4 Thus, a directed line segment has magnitude as well 1 A quantity that has magnitude as well as direction is called a that a directed line segment is a VECTOR (Fig (iii)), denoted as ABuuur orsimply as ar, and read as VECTOR ABuuur or VECTOR ar .The point A from where the VECTOR ABuuur starts is called its initial point, and thepoint B where it ends is called its terminal point. The distance between initial andterminal points of a VECTOR is called the magnitude (or length) of the VECTOR , denoted as|ABuuur|, or |ar|, or a.

5 The arrow indicates the direction of the VECTOR .$Note Since the length is never negative, the notation |ar| < 0 has no VectorFrom Class XI, recall the three dimensional right handed rectangular coordinatesystem (Fig (i)). Consider a point P in space, having coordinates (x, y, z) withrespect to the origin O (0, 0, 0). Then, the VECTOR OPuuur having O and P as its initial andterminal points, respectively, is called the position VECTOR of the point P with respectto O. Using distance formula (from Class XI), the magnitude of OPuuu r (or rr) is given by|OP|uuur =222xyz++In practice, the position vectors of points A, B, C, etc.

6 , with respect to the origin Oare denoted by ar, ,bcrr, etc., respectively (Fig (ii)).Fig NCERTnot to be republished MATHEMATICS426 AOP90 XYZXAOBP()x,y,zCP()x,y,zrxyzDirection CosinesConsider the position VECTOR or OPuuurrr of a point P(x, y, z) as in Fig The angles , , made by the VECTOR rr with the positive directions of x, y and z-axes respectively,are called its direction angles. The cosine values of these angles, , cos , cos andcos are called direction cosines of the VECTOR rr, and usually denoted by l, m and n, Fig , one may note that the triangle OAP is right angled, and in it, wehave ()cos stands for | |xrrr =r.

7 Similarly, from the right angled triangles OBP andOCP, we may write cos and cosyzrr = =. Thus, the coordinates of the point P mayalso be expressed as (lr, mr,nr). The numbers lr, mr and nr, proportional to the directioncosines are called as direction ratios of VECTOR rr, and denoted as a, b and c, NCERTnot to be republishedVECTOR ALGEBRA427$Note One may note that l2 + m2 + n2 = 1 but a2 + b2 + c2 1, in Types of VectorsZero VECTOR A VECTOR whose initial and terminal points coincide, is called a zerovector (or null VECTOR ), and denoted as 0r.

8 Zero VECTOR can not be assigned a definitedirection as it has zero magnitude. Or, alternatively otherwise, it may be regarded ashaving any direction. The vectors AA, BBuuur uuu r represent the zero VECTOR ,Unit VECTOR A VECTOR whose magnitude is unity ( , 1 unit) is called a unit VECTOR . Theunit VECTOR in the direction of a given VECTOR ar is denoted by Vectors Two or more vectors having the same initial point are called Vectors Two or more vectors are said to be collinear if they are parallel tothe same line, irrespective of their magnitudes and Vectors Two vectors and abrr are said to be equal, if they have the samemagnitude and direction regardless of the positions of their initial points.

9 And writtenas = of a VECTOR A VECTOR whose magnitude is the same as that of a given VECTOR (say, ABuuur), but direction is opposite to that of it, is called negative of the given example, VECTOR BAuuur is negative of the VECTOR ABuuur, and written as BAAB= uuur The vectors defined above are such that any of them may be subject to itsparallel displacement without changing its magnitude and direction. Such vectors arecalled free vectors. Throughout this chapter, we will be dealing with free vectors 1 Represent graphically a displacementof 40 km, 30 west of The VECTOR OPuuur represents the requireddisplacement (Fig ).

10 Example 2 Classify the following measures asscalars and vectors.(i) 5 seconds(ii)1000 cm3 Fig NCERTnot to be republished MATHEMATICS428 Fig (iii) 10 Newton(iv) 30 km/hr(v) 10 g/cm3(vi) 20 m/s towards northSolution(i) Time-scalar(ii) Volume-scalar(iii)Force- VECTOR (iv)Speed- scalar(v)Density-scalar(vi) Velocity-vectorExample 3 In Fig , which of the vectors are:(i) Collinear(ii)Equal(iii)CoinitialSolution (i) Collinear vectors : ,andacdrrr.(ii)Equal vectors : (iii)Coinitial vectors : , graphically a displacement of 40 km, 30 east of the following measures as scalars and vectors.