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12. Linear Combination of Vectors, 18. Properties of Vector Addition, If a vector r can be expressed in terms of other, vectors a, b, c..a r- xa + yb + zc +.., where x, y,, (ii) a+(5+c) = (a+ 5) +č, z are scalars, then r is called linear combination of, a, 6, c., () a+(-a) - 0, 13. Collinear Vectors, 19. Addition of more than Two Vectors, If a mymber of yectors are represented in magnitude ar, direction by the sides of a polygon taken in order the, the sum or resultant of these vectors is represented b, the closing side of a polygon in the opposite direction, If vectors a and b are collinear, theri either of the vectors, a or b can be expressed in terms of the other vectors, Le., an xb, 20. .Şybiraction of Two Vectors, or, If. .a and b are two vectors, then the vectul, 14. Localised and Free Vectors, a vbma+ (-5) represents the subtraction operation, A directed lino segment drawn parallel to the direction, of a given vector through a specified point in space is, called a locallsed vector. Obviously, there can be only, one such vector., To subtract vector b from vector a, we add negative of, vector b. to the vector a., 21. Multiplication of a Vector by a Non-Zero, Scalar, A directed line segment drawn parallel to the direction, of a given vector is called a free vector. In this case the, origin or the initial point is not specified., If m is a scalar and a is defined as a vector whose, 15. Coplanar Vector, magnitude is m times the magnitude of the vector a, The vectors lying in the same or the parallel planes are, called coplanar vectors:, and whose. direction is tha: of the or oppositle, vector a, to the direction of the vector a, according as m is, posilive or negative, 22. Properties, 16. Negative Vector, The vector which has the same magnitude as a vector, a but opposite direction, is called negativo of a and is, a and, If m and n aro two non-zero scalars and a, two vectors, thcn, denoted by (0, - a)., are, 17. Addition of Vectors, (0 m(na) = (mn)a = n(ma), 1. Trlangle Law of Vector Addition. If two vectors, are represenied in magnitude and direction by two, sides of a triangle then resultant of these two, vectors is represented in magnitude and direction, by the third side of the triangle taken in osposite, (ii) (m + n)a = ma t na, (iii) mla + b) = ma + mib, 23. Methods to test the Collinearity and, Coplanarity of Points, () Collinearity of three points, order., 2. Parallelogram Law of Vector Addition. If two, vectors are represented in magnítude and direction, by adjacent sides of a paralielogram, then their, Kum in magnitude and direction is represented by, the diagonal through the point of intersection of, these vectors, in the opposite direction., Let A(a), B(b). CC) e three points; then find, the vectors AB and C and if, 1

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AB = ,BC, Conversely, if a.b=0 then 0 = 90°. i.e.., then points A. B. C are co.l near., vectors a and b are perpendicular to each, Alternatively, :0 establish the collinearity of, other., A(ai, B(è). ard (2) the following results can, If two vectors are equal, then, be used., a.a-la|lalcos 0- lap pa', it is alsc, written as a'. Thus square of a vector a i!, a scalar which equals the square of it, magnitude., If xa + yb + zc = 0 such thai 1, y. z are not all, zero, the: the points A, B. C are collinear., (ii Coplanarity of Three Vectors. Three vectors, a.b.i are coplanar, if ore of thèse cañ be, Éspressed in terras .of the other, i.eit, (ii) In case of mutually perpendiculat uni, vectors, i, j, k we have, ii - j.j = kk = 1, or, (iin Coplaoarity of Four Polnts. To ptove ihe, and, 1.j= jk = ki = 0, coplanzrity of four points Ala), B(5), Cé)., D(d). find the AB, AC and AD and express one, If, a-a,i + azj+ ask and, 5=b」+らj+らk, then, of these vecters in tems of the other two as shown, above Aliernatively, these four poinus are coplanar, if the following results.are true., ab- (a,l +azj+ ajk). (i + bi j + byk), -ab, + agb, + azby., (a) xa + yb + ze + rd -o, Note. The Scalar Product of vectors i, (b) 1+ y- +I= 0, wbere x, y. z. are not, distributive. L.e. a(b + c) -a.b+a.c, zll zero a uhe same time., (un Geometrical Interpretation. Lot OB =, 24. Product of Two Vectors, Vectors can be muluplicd in two waya and hence there, zre two types of products. Scalar or Dot product and, and OB -b and e the angle between th, vectors. Then, we know, Vocior or Cross product. If a and 6 are two vectors,, then a, b and a xb are called dot and cross products, =lä|05|cos 0), "espectively., (a) () Scalar Product of Two Vectors., 'ā' rojection of b on OR), The scalar or do: prodact of two vectors a, and o is definod as the scalar abcos 0,, where a is the angle between the vectors., Thus, a, b =la||b|cos:0, If twa vectorg are perpendicular to eaách, othe:, 1.e. 0-90°., .A, Also,, then, a.6-0, -5çä cos 8), Also, -(Projection of a on OB), 2

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(b) () Vector Product of Two Vectors, = (Czb, - asb,)i + (azb, - a,b3)j, The vector or the cross-product .. two, %3B, + (a,b, - azb, )k, vectors, a and b is defined as the vector, (H) Geometric Interpretation. Let the vectors, la||b| sin e fi. where e is the angle, be represented by OA and OB, respectively. If fA is a unit normal vector to, and, between the vectors and f is unit normal, vector-to the planc containing a and b., Thus, the plane of the AOAB, then, x%3D, -ab sin 0 n, The direction of: the vector axb is, %3D, perpendicular to the plane in which a and, 6 lie., = AOÁB = 2A OAB, If the two vectors are parallel, then e 0°, or 180°, axb=la||5|sin e n, A, Conversely, if axb=0, then e= 0 or, Hence axb, of the triangle formed by the adjacent sides, represented by the vectors a and b., 180°. I.e. vectors a and b are parallel to, represents twice the vector area, cach other., If two vectors are equal, then axa = 0., Also axb can be interpreted as the vcctor, Also, axb=-bxā, area of the parallelogram with a and b as, the adjacent sides., (li) In case of mutually perpendicular unit, vectors i, j, k we have, ? 25. Multiple Products, ixi = jx j=k xk = 0, Let a, b. c be the three vectors, then. since axb itself, is vecter, it can be multiplied by č, scalarly as well as, voctorially, forming scalar and vector triple products., (1) Scaler Triple Product. The scalar or the dot, Also,, ixj=k, jxk - i, k xi = j, and, jxi =-k, kxj =-1,ixk = -j, Note. The vector product of two vectors is, disributive i.e., product of ax5 with the vector c is defined as, ax (b + c) = axb+ axc (b + c), .* thb scalar produc: of the three vectors or scalar, (U) The vector product of two vectors can be, expressed in the determinant form as below., triplo product and is writteo as (axb), c or, Let a = a,i +az)+ a,k and, (a,b.c)., (if) Geometrical Interpretatlon. Let the vectors,, b = b,i + bj + bzk. Then, a,5,c be represented by OA, OB End OC, respectivėly. Construct a parallelopiped with OA,, OB and OC as the coterminus odges. Let V be ils, vclume., az aj, 3

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a.6.c, (i) If any two of the vectors a,, are equal) then, C-, labc)=0., (iii) Every scalar triple product say, axb.c, is, independent of position of dot or dross, becanse, 2,, of, axb.c=bxc.a =cxa.b,, because the above products represent the volumo, of the same parallčlopiped., Now, laxbl= arca of the parallelogram, OADB = a (say)., (iv) Analytical expression for scalar triple product Let;, Also, |OC = OC = AD =c, AL = c cos q = p., a = a,i +a j+ azk,b = b,i + b, j+ b,k, Then (axb)c = laxb||clcos 0, = a cos 0 = volume of parallelopiped V, and c= ci +c,j+ c,k then, Then (a xb)c, represents tho volume of, (abc) =|b by by, paralleolopiped, with ab and e, as coterminus, cdges., Thus, the scalar triple product, 27. Vector Triple Product, (1x j)k = k.k = 1., It is obvious that the volume of a parallelopiped, (1) The vector or the cross product of a xb with the, vecior c is defined as the vector product of the, three vectors triple product, written as, having a, b.c as coterminous cdges is same as, (axb) xc, the volume of paralleopiped having-h,e, a or, Note. In this case the presence of the brackets is a, must, because, c, a. b as coteminous edges. Thus, ax (bxc)* (ax b)xc, (ii) Expansion Formula. The vector triple product can, be expressed as, (axb)xc= (c.a)b- (c.b)a, Sometimes these results aro wriuen without use of, bracket, ie.,, axbxc, bxc.a and exa.b, The removal of brackets causCs no umbiguity,, fic., (Extreme x inner)x Outer, hecause ax (bxc) is meaningles. Studerits are, - (Outer. Extreme) inner, however advised to uso brackets., - (Outer . inner) extrome), (ii) Volume of tetrahedren with edges as a, b.c, Similarly, a x (b xc) = (a.c) b – (a . b)c, ,28. Applications of Vectors, (a) Geometricnl, 26. Properties of Scalar Triple Products, () If threc vectors ā, 6,7 are coplanat, then (a, b, c], (1) Vector equation of a Straight Line,, The equation of a straight line passing, through a given point A(a) and paralled u, e vectór bis given by, i.e. a.(bxc) is zero, because no-parallelopiped, can be formcd with these vectors as coerminus, edges. Thus the condition for three vectors to bc, Foplanaris that, (r-a)xb =0,, whore r. is the position vector of any pop, on the given line. The above cquation is th, (abe) =0, 4

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VECTOR ALOEBRA, vector equation of the straight line in., nonparametric form., BA x F.", Thus the moment of, The parametric equatlon af tho above lịne is, a force about a point, is a vector, its, tb., B, đirection being, tó the plane of the, (ii) The vector equation of straight line passing, through the point A(a) and B{b) is given, by, line AB and the, force F., raa+ s(b - a) + !(c.- a), 30. Reciprocal System of Vectors, = (1-s-1)a+sb+ic., Let 9,6, c be the given vectors such that (a b c) 0,, The vector equation of a plane passing, through two given' poínts_A[a), B(b) and, porallel to the vector c is given by, then, •bxc, E cxa, d =, (abc), r=a+s (b-a) + 1(c - a), (abc), = (1-s-1)a + sb + ic., (lin Rado Formyla. The position vector of a point, R(r), which divides the line joining the points, (abc), are called reciprocal vectors of ā, 6.7 respectively, B(b), Properties of reclprocal vectors, A(a) and B(b) in the ratlo m:n is givek by, n+b+c+..., cler b.2-0, č.B-0 a.2-0 (::ā. (āxb) = 0), Thus position vector of the centroid OG)of, the triangle whose vertices are A(a), B(b), | = 1, and C(c) is given by, atb+c, 3, 31. Prpducts of Four Vectors, 29. Work done by a Constant Force, 1. Scalar Product of Four Vectors a,b.c.d is, 1' Let F be a constant force, which while acting on, particle, displaces it from the point A lo the point B., defined as, The work done, (axb). (exd), by a constant, force is the, a.c a.d, Also, (ax3). (exa)-, scalarproduct of, force (vector), and, the, ,2. Vector Product of Four Yectors, displacement. Jt, Is quile obvious if force acts perpendicular to the, direction of the displacement, the work donc is, a, b, c,d is defined as, zero., 2 Moment of a Force about a point, Let a constant force F act at a point A. Then, moment of F about the point B is given by