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Chapter, 10, , Veetor Algebra, r i l a b u s :, , scalars, magnitude and direction of a, vector, cosinestrati of vectors., Types of vectors, Zero, parallel and colinear, vector of a point, negative of a vectors),, vector, nents of a vecto, addition of yectors,, coeation of a vector by a scalar,, kctorsandsc, , Dinecion, , (qual unit, zero, , xSTiOD, , nt dividing, , line, , segment in given ratio. Scalar, , a, , Vector, , u, , c, , Vectors: (ii) and (vi), , a, , roduct of vectors, projection of a vector ona, dot), (cross) product of vectors, scalar triple, ne, , vectors. ) 5 seconds (ii) 1000 cm (ii) 10, Newton (iv) 30 km/hr (v) 10 g/cm° (vi) 20 m/s, towards south., Ans: Scalars: (i), (i), (iv) and (v), , position vector of, , 7ulipdi, , a, , 5. Classify the following measures as scalar and, , t, , 6. Represent graphically a displacement of, , 40 km, 30° west of south., Ans:, N, , Exam c o r n e r :, , Marks, , 12, , No.of, , W, , marks, , 2 2, , qns, , Total 11, , 3, , 30°, , 5 questions, , 1. Define a scalar. Give example., Ans: A quantity that has magnitude only is known as, a scalar. For example: Mass, length, time,, temperature, density etc., , 2. Define a vector. Give example., , 7. Represent graphicaly a displacement of, 40 km, 30° east of south., Ans:, , Ans: A quantity that has magnitude as well as, , N, , direction is called a vector., , For example: Force, velocity, acceleration,, momentum etc., , .Classify the following, , measures, , as scalars and, , 40 km, , vector i) 10 kg (ii) 2 meters north-west, 10" coulomb (vi) 20, 40°, 40 watt, , ii), , (iv), , B0, , (v), , m/s., Ans:, , Scalars: (i). (ii), (iv) and (v), , W, , Vectors: (i) and (vi), and vector, Classify the following as scalardistance, (ii), quantities. () time period (ii), force (iv) velocity (v) work done, , Ans:, , Scalars: (), (i) and (), Vectors: (ii) and (iv), , Student's ilkuminator, , S, , 8. Define the magnitude of the vector., Ans: If a = AB is a vector, then the magnitude of the, , vector is ll or 1ABE AB., , I PU Mathematics

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336, , Vector Algetr, Define the following:, () Zero vector (i) Unit vector, , Equal, , vectors (iv), , Negative, , Here PQ IAB QP and AB are unlike, , (2014J) (i), , of, , vectors., , vector, , (ix) Free vectors: If the initial point of, of aa vecte, vector, not specified then it is said to be a free vector is, , vectors (vii) Unlike vectors, (ix) Free, vectors (x) Localised, vectors (xi) Coplanar, vectors (xii) Position vector of a, point., Ans: () Zero vector:, A vector whose initial and, terminal points coincide is called a zero, vector,, , (x)Localised vectors: A vector drawn parallel to, given vector through a specified point as, the, initial point is called a localised vector., , 2015M)V), , a, , Coinitial vectors (vi) Collinear, vectors (vi) Collinear, vectors (2017M) (vii), Like, , denoted by O., , ector, , (xi) Coplanar vectors: Three or more nonz, same, , Note:, , The magnitude of, a zero vector is 0 but it, cannot be assigned a definite direction., () Unit vector: A vector ä is, called a unit vector, if la=l and is, à, , denoted by, , lying in the same plane or parallel to the, plane are said to be coplanar, otherwie, wise, , vectors, , they are called non-coplanar., , (xii) Position vector of a point:, , Let O be the, origin, such that OA, =, then, , and let A be a point, say that the position vector of A is ., , we, , (ii) Equal vectors: Two, , vectors, and b are said., be equal if, they have the same magnitude and, the same direction, regardless of the positions of, to, , their initial points., , a, , Then d=b, , (iv) Negative of a vector: A vector, having the, , magnitude as that of a given vector, direction opposite to that of a is, , negative of, , and denoted by -ä, , (v) Coinitial vectors: Two, the, , same, , initial, , or more, , same, , and the, called the, , vectors, , 10. In the figure, which of the, vectors are, (i) collinear (i) equal vectors (ii) coinitial, vectors?, , having, , C, , point are called coinitial vectors., , O, , >B, , d, , A, , Then, , OA, OB, , and OC, , co-initial vectors., , are, , (vi) Collinear vectors: Vectors having, , the, , same or, , parallel supports are known as collinear vectors., Here, , AB,, , BC and AC, , are, , collinear vectors., , Ans:, , () Collinear vectors: ä, and d, i) Equal vectors:, and, , (i) Co-initial vectors: b,, 11. In the, , and d, , figure, identify the following vectors, , (vii) Like vectors: Collinear vectors having the same, direction are called like vectors., (vii) Unlike vectors: Collinear, , vectors, , opposite directions are known as unlike, A, , P, , Student's illuminator, , having, , d, , b, , vectors., , B, , Q, , () Coinitial, , (i) Equal, , (ili) Collinear but not, , equal., I, , PU Mathematics

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eVan, , lcatiobn, , ons, and, , a, , d, , 15. Prove, , Coinitia:, , (i), , Equal:, , b, , and, , i ) Collincar, , Ans: Let, , d, , as, , true, , or, , -ã are collinear., , ), , =, , tors, , collinear, , BC=, , C, , false., , True, , and, , Two, , AB=b and, , OA ,, , but not equal: ä and, , Answer, the following, wer the, (i) a, , is associative., that vector addition, , are, , False, , always equal, , in, , m a g n i t u d e ., , ( i ) Two, , vectors having same magnitude are, , collinear. False, , Two collin, , vectors having the same, False, magnitude are equal., , (iv), , 13.Write the, ns:, , In, , triangle, , AOAB,, , a, , OB=atb, , law of vector, , additio, , if OA =d and AB, , This is known, , as, , the, , =, , b, , a, , Join OB, AC and OC., , (+b)+, then, , triangle, , of vector addition., , law, , =, , AB) + BC, , (O, , =OB +BC, , =, , ..(1), , OC, , a+(b +)= OA +(AB+ BC), ..(2), , =OA+ AC =OC, From (1) and (2) we get, , (ä+b)+, , =+, , (b +), , b, , 16. Show that, , Þ is, , the additive identity for, , vectors., , Ans: Let, , 0, , and, , 14., , addition is commutative., , Prove that vector, , Ans: Let OA =, , and, , AB =b., , Complete the parallelogram, , OA= ä., O+ =, , Then, , AA, , ä+Ö=O, , AA+OA =OA, , =, , =, , OA, , =, , ä, , ä, , +0=0+ =, , 0 is called the additive identity for vectors., , OABC., , 17. Show that there is, , C, , an, , additive inverse for, , every vectors., , Ans: Let OA=ä. Then A0= -, , +(-ä)=OA+, and (-)+, , For every vector ä, there is (-a), which is, additive inverse of ä., , Then OC=AB=b, , CB OA=, , 18. Define direction ratios of a vector., , + b =OA+ AB = OB, , ...(), , and b+ä=O +CB=OB, , .(2), , ätb =b+d., Student's iurninator, , = A0 +OA = AA =Ö, , Hence, ã +(-ä) =(- )+ä=ô, , a, , From (1) and (2),, , AO =00= ö, , we, , get, , Ans: Let r = ai +bj + ck be any vector. Then a, b, c, , are called the direction ratios of F., 19. Define direction cosines ofa vector., Ans: If r=ai, , +bj +ck, , is any vector then, , PU Mathme tics

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Vector Aigebr, , 338, b, , C, , va +b +c va' +b+? a +b+, , are, , called the direction cosines of, f, , Ans:, , P, , magnitude, , P(x,,y,z,), , of, , and, , (x, - x,}i +(y, ->)j +(-)k, , Scalar components, , (x,-x). (;-i», , are, , (z2), , 27, If vector, , Compute, , AB =, , 2i-j +k, , and, , OB 3i-4j+4k , find the position vector, , (2016M), , OA., Ans: We have AB= 0B, , OA, , -, , ., , OA= OB-AB, , -i-4j+48)-(zi -j+k), , IPQ=x- x, +(y- y,} +(2,-z,, 21., , sum, , Ans: ä +b+iz-4j-k, , 20. Find the scalar, components and, the vector, joining the, , points, , of the vectors =i-2/+i, 6= -2i+4j + 5k, =i-6j -7k, , 26. Find the, , the, , vectors. (i), , magnitude, , ä=i+j+k, , of the, , (i), , =i-3j+3k., , following, , b=2i -7j -3k, , 28) Find the unit vector in the direction of the, , (20173, 2018M), , vector a =i +j+ 2k, , Ans: 6) 1áE1+1+1=3, , i) lbEV4 +49 +9 = 62, , (ii) 1, , Ans: The unit vector in the direction of, , i+j+2, , l l 1+1+4 6, , 29, Find the unit vector in the direction of the, vector d = 2i +3j +k, , 22. Write two different vectors having same, , magnitude., Ans: Let d, , i+j+2k), , =itj+k and b=i-j+k, , (2015J), , Ans: The unit vector in the direction of, , d a 2i+3j+k-(2+3j+k), V4+9+, 2i+3+k), lal, 30. Find the unit vector in the direction of the, , 23. Write two different vectors having same, direction., , Ans: Let ä=i+jtk and 2ä 2i+2j +2k Then, and 2 having same direction., , 24. Find the values of r andy so that the vectors, 2i+3j and xi + yi are equal., , Ans: Given 2i +3j, , =, , xi+yj, , Equating the co-efficients of i and j, we get, , vector, , point (-5, 7)., Ans: Let A (2, 1) and B (-5, 7) be given points., , AB =OB-0A=-7i+6., , P, , (1, 2, 3) and Q (4, 5, 61./, , OP =i+ 2j +3k, , Ans: Given, , and, , O-4i+5)j+6, , PO-00-OP=3i+3j +3X, 31. Find the unit vector in the direction of ä+b,, , where, , 2i -j+2k, , and, , b, , =, , -i +j-k, , Ans: Here +b=i+k, The unit vector in the direction of ä +b, , x= 2 and y=3., 25. Find the scalar and vector components of, vector with initial point (2, 1) and terminal, , PQ where, , +b, , 32. Find a vector in the direction of vector, , 5i-j+2k which has magnitude 8 units., Ans: Let, , =, , 5i -j+2, , Scalar components are -7, 6 and vector, , components are -7i, 6j., Student's itluminator, , I, , PU Mathematcs

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339, j e e v i t hP u b l i c a t i o n s, , S i - jin, + the, 2, roCtion., direction, of ä, vector, , a-, , unit, , A, , si-+2, i-+2), a1 25+1+4, vector, , Rcquired, 33,, Find the, 13,Find, , ., , Then nAP = mPB, , PB, , j+2k)., , = 8a 30i, , m, , of the vector, , direction cosine, , i+2j+3k.(2014M), l+2j+3k EVI+4+9= 14, , ns:, , B, , Direction, , cosines of the, , 2, , given, , vector are, , nAP =mPB, , 3, , M4'1414, 4. Find the, , n(OP-OA) =m(OB-OP), , direction cosines of the vector joining, , (m+n) OP, , the points A (1, 2, -3) and B (-1, -2, 1), , directed, , from A to B., , =, , mOB + nOA, , mb+ n, , F =, , m+n, , AB OB-OA -2i-4j +4k, =, , Ans:, , which, 37. Find the position vector of a point R, divides the line joining two points P and Q, , ABEV4+16+16 6, , AB, , The direction cosines of, , whose, , are, , position, , +j+k, , are, , i+2j-k, , the_ ratio, , respectively, in, , and, , 2:, , 1, , (2017M), , () internally (i) externally., , i.e.,, , Ans: Let OP =i+2j-k and OQ=-i+j+k, , 35. Show that the vector, , i+j+k, , is equally, , inclined to the axes Ox, OY and 0Z., , Ans: Let ä=i+j+k., Let vector, , vectors, , make angles a, B. y with the axes, , Ox, oY and OZ, then their direction cosines, , (i) Internally OR, , OP+200-i+4j+, , =, , (i) Externally OR=, , 1+2, , 3, , +200_-3i+3k, , 2-1, , are, , cos o, cos B, cos yY., , 38. Consider two points P and Q with position, , vectors OP =3ä -2b and 00= +b., , Now, , Find, , the position vector of a point R which divides, , +1+1, , COs, , c o s a s cos ß = cos y, , Hence, the given, , vector, , COSy, , the line joining P and Q in the ratio 2: 1., )internally and (i) externally., , a=ß=y, , is, , equally, , inclined to, , Ans: Internal division: OR=200+1 0P, 2+1, , axes., , the formula for position vector of a, point which divides the join of two points, , 36/Derive, , A and B internally in the ratiom:n., , (2015M, 2016J, 2017M, 2018M), Ans: Let O be the origin., Then OA =ä and OB =b, Let P be a point on AB such that, , Student's iluminator, , External division OR =00-1OP = 452-1, 39. Find the position vector of the mid, point of the, vector joining the points P (2, 3, 4) and, , Q4,1,-2)., , Ans: Let, , OP =2i+3j+4k, o0=4î+i-2, l PU Matr ama tics

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340, , Nector AiGet, Mid point of the, , vector joining the, , points P and Q, , OP+00 2+3+ 4k)+(4i ti-2), 2, , a, 43. Write down, an, , angle 30°, , the, , with, , plane, ane, n, , positive, , X-axis., that, Ans: We know, , =i+2+, , XY, , 40. Show that the points, , unit vector in XY, , is the unit, , cosi+ sin 8j, , plane. Here 0, , =, , directionvakinofg, , 30°, , 30°i +sin 30j, , Therefore required vector = cos 30°i + sin 20e, , A(ü-j+k),, , B-3-5, c(si-4-4X) are the, vertices of a right angled triangle., , Ans: Given OA =2i -j+k, OB =i-3j -5k, , 44.A girl walks 4 km towards west, then, , walks 3 km in a direction 30° east of north. she, , OC=i-4-4k, , the girl's, stops. Determine, , AB OB-OA=-i-2j-6k, BC OC-0B =2 -j+k, , displacement from, , her initial point of departure., IABE3 km, Ans: Let IOAE4 km,, , AC CA=OC-OA=-i+3j+SX, AB AB=1+4+36=41, B, , BC BC P=4+1+1=6, , CA =CA, BC+CA, A, , =1+9+25 =35, , =AB?, , 30, , ABC is a right angled triangle., , 41. Show that the points A, B and C with position, , M, , vector ä 3-4-4k, b =2i-j+k and, -i-3j-5k, respectively form thevertices, , of a right angled triangle., Ans:, , (Similar to 38, try yoursel), , OAB=90°-30°= 60°, Let OB-xi+ yi, , 42. Write all the unit vectors in XY - plane., , Ans:, , Ix=OM OA-AM, , Y, , 4-1ABIcos8 =4-, , P,), , ly BM =ABsin60°=, X, , 10B-P-2-n, Let OP xi + yj be the unit vector in XY., , plane. 1OP+y=1, From the figure, x = cos 6 and y = sin 0, , Unit vectors in XY, , plane are cos ei +sin 8j, , 45. If, , i+j+k, i+5j, 3+2-3k, , and, , i-6-k are the position vectors of points A,, B, C and D respectively, then find, the ang, between AB and CD. Deduce that AB and, CD are collinear., Ans: AB, , of A, , =, , position, , vector, , of B, , - position ve, , =(2+5)-+j+k)=i+4j-k, Student's illuminator, , I, , PU Mathematics

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341, , 1 e e v i t hP u b l i c a t i o n s, , CD OD-oC.=i-6-X)-(i+2-3X), , Observations, , a-b is areal number, , -2-8+2k, , - -2(+4j-k)=-2AB, , Let a, , CD and AB are unlike vectors and hence, , a b, , they are collinear., , between two AB and, , Angle, , 2i, , 6Show that the vectors, , CD =180°, , -3 +4, , and, , + 6 - s k are collinear. (2015J), a, Ans: Let, , =, , 2i-3j + 4k, , and, , 4i+6j-8k = -2(2i-3j +4k), , and b be, 0, , iff, , a and b, , are, , 2i, 47. 1If the vectors, , aa, , If 0 T, then a b=-lallbl, , -, , a -a--1a, , R=Ri=0, , ii-1:=RR=1, The angle between two, , non zero vectors, , a-b, l llbl, , The scalar product is commutative. i.e.,, , collinear., , +3j-6k, , are, , I f 0 0, then d b llb, , -2, b, , zero vectors, then, perpendicular to eacht, , non, , other, , and b is cos, and, , two, , and, , 4i -mj -12, , a b=b, a-b+)=a-b+b c, , are parallel, find m., Ans: Given vectors are, , parallel, then their, , components are proportional. arr, , aa):b=a b)=ä ab), Scalar product in components:, , Let aai+aj+a,k and b=bitb,i+bk, , m=-6, , Then a b, , ah tab tab, , 48. Show that the points A(-2i +3j + 5k),, , Bi+2j +3k), , and C(7i -k), , are, , collinear, , Ans: AB = OB - OA = 3i-j-2k, , Projection of b on, , a-4b, , The projection of ä on b-40, 1b1, , AC OC-OA, , =9, 3j-6 =3(3i-j-2X)=3AB, ACIAB., , 50. Find the angle between two vectors, with, , Therefore A, B, C, , are, , collinear, , 49. Define scalar product of two vectors., , magnitudes v3, , and 2,, , and b, , respectively,, , having ã 5= 6., Ans: We have cos e=4b, , Ans: Let ä and b be two vectors and 0 be the angle, between ä, , and b, , Then, the scalar or dot, , is, b, denoted by :b,, defined by ä-b «äWbicos6 where 0 0sn., , product of, , . cos= 2, , 0-, , and, , b, , 51. Find the, , angle between, and 3i 2j+k., , Ans: cos=, , the vectors, , J+3k) (3-2+k), , +-2j +3k, 10, , 1+4+9 9+4+1, o, , o, , a, , Student's ituminator, , cos, I PU Mathematics

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344, , Vector Aigeta, , and lä=81b1, , ä t +lbP -2ä b, , .(2), , From (1) and (2), , we, , = 4+9-24) = 5, , 64ibP -1b P=8, , get, , lã-bE5., , 63, , V63, , 70. If ä,6, are unit vectors such that, a+b+ë = 0, find the value of, , Now lä8, 2216, 63, , V63, , 67 If ä +b=, , -6,prove that ä and b are, , ä+6., , + ä., , Ans: Given, , +b + =0, , Squaring on both sides, we get,, , perpendicular. (2016M), , a, , Ans: Given lä+bl=-b, , +16P +IP +2(äb +b +.ä)=0, , Squaring on both sides we get,, , 1+1+1+2(ä-b+b-+, , la 6+2a-b =|at +6 -2ä-6, , ä-+6.+-ä=, , ä-b =0, a, , andb, , )=0, , 71) Three vectors , b and, , are perpendicular., , 68) Find the projection of the vector, a=2i +3j +2 on the vector b, , condition ä +b+, , =0. Evaluate the, , -ä.b +b-+=i, , +2j +k., (2014), , satisfy the, quantity, , iflä=1, 1bl=4 and, , (2017M), Ans: We have l +b + f4ät, , +1bP +1f+, , 2(a-b+b +ë-), , Ans: The projection of ã on b=, , bl, , 0=1+16+4+ 24, , =2+6+2 10, Vi+4+1, , 6, , () i-j on i+j, , 72. Let ä,b and, , Ans: The projection of, , on b=d*, Ibl, , 1-, , Ans: Given l(b+ )=i(b+ )=0, , ii+3+7Tk on Ti-j+8k (2015/,2018M), Ans: The projection of ä on b=a, 60, , 49+1+64, , 14, , 69 Find l-b1,, , a-b+ä., , =0, , ...(1), , bL(+ä)=b-(+)=0, b.+b-=0, , .2), , EL(+b)=i-(ä+b)=0, , if two vectors ã and b are such, , that la=2, 16=3 and ä-b =4., (2014J, 2016J), Ans: Given l =2,1b-3 and ä-b=4, , Consider 1-6P=(-b), , la-3, lb l=4, 1k5 and each one of them, being perpendicular to the sum of the other, , two, find lä+b +1., , =0, , 7-3+56, , be three vectors such, , + .b, =0, (1)+(2) + (3) gives, , ..3), , 2(-b+b.+-ä)=0.4), lä+b+t, , (ä-b), , = 9+16+25+0, Student's iluminator, , P U Mathematics

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350, , Vectr Age, For, , example,, , parallel, , let ä and b be the two, non-zero, vectors, then dxb 0., , i-gxk)+j-ixk)+k-(ixj), , =, , This shows that converse, of the, is not true., , given, , statement, , Ans: i-jxk)+j(ixk)+k-Gx), =ii+j-)+kk =1-1+1=1, , 97. Let the vectors, , ä,b,, , be, , given as, , a,i+a,j+a,k, bi +b,j +b,k,, , 101., , b+= (6, +c,} +(6,, , Ans:, , +c)j +(,+G,)k, j, , i, , k, , ax(b+)=, t, , Iflä-b lxbl, , then find the angle, , between, , a and b, , i+cj +c,k Then show that, äx(b +) =(xb)+(a x)., Ans:, , 100. Find the value of, , b+C b+c|, , 1dbHäxbl =lllbl cos6 dlbisn8, tan 0=1 0 -, , 102. Define scalar triple product., Ans: Let ,, , b and, , be any three vectors. Then, the, , scalarproduct of ä and (bx), , ie.,, , (bx) a, , called the scalar triple product of ä, b and ., , Observations:, a-(bx) is a scalar quantity.(color t r, = ( xb)+, , 98, Let the vectors, , 5, , and, , (x), , and b be such that l=3,, , lx6=1., , Find, , the angle, , between a and b (2015J), Ans: We have lxb, , l llbi, , sin 0, , The magnitude of the scalar triple product is the, volume of parallelopiped fomed by, adjacent, , sides given by the three vectors a, b and, , Volume= (bxa, , Ifä=aitaj+4, b=hi+bj+bksnd, C=GitajtGk then, , l=3sin0, sin 6=, 99. Find the area of the rectangle, whose vertices, , ci-.nd oi-j.a., , If ä,b and, , be any three vectors, then, , a (bx)=(äxb)., , Ans: AB=OB-OA 2, , AD, , 0D-OA, , =, , -j, , Area of the rectangle ABCD, , Three vectors a, b and c are coplanar ift, , ABU ADIsin=2x1xl=2, , (bx)=0, , Student's ilurminator, , I, , PU Mathematos

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J e e v i t h, , 351, , P u b l i c a t i o n s, , ä-(x ), , if G) ä =2i +j+3k,, , 20-6-4+-6=0, , 3 .Find, , i + 2 j + k and, , =3i +j+2, , =, , 21, 31, 2, , i) a=ai-10j-5k. 5=, , ã-(b x ) =|-1, , Ans:, , and, , Ans: Given vectors are coplanar., , = 2(4-1)-1(-2-3) +3(-1-6), = 6+5-21 = -, , -10, , -7, , 10, , i-2j+ 3k, B=2i-3j+k and, 3 i +j-2, , -5, , -5, , 0=0, , -3, , 1, , (15-0) +10(21-0) -5(28+5), , 0, , =, , 15 +210165 = 0, , 2i-3, 5=i+j+k and =3i-k, (Try yoursel) ii) 24 ii), , =-3, i), , -14, , 104. Find the volume of the parallelepiped whose, , coterminous edges are represented by the, , =ai-j+ak, 1, , and =i -3+5k., , 2, , 1-2 -3, 3, , ä -i-j+k, 6 -2i+j-k and, , Ans: Given vectors are coplanar., , vectors ) a =i-2j -3k, b = -2i +3-4R, , Ans: ä-(bx)=|-2, , -7i-5j, , c=i-41-3k, , 31 2, , Ans:, , 2, , -1, 1, , 1, -1=0, , i(a-1)+1 (22, , 4, , +2)+1-2-.) =0, , -1+3-2-A=0, =1, , 1-3 s, = 1(15-12) +2(-10+4)-3(6-3), , G) ä =i+3j +k, i =2i-j-k and, , 3-12-9=-18, , i+7j+3k+3k (2015J), , The volume = l -(bx )=18, , Ans:, 105., , Ans: Given, , coplanar., , (i) d-i+3+k, b = 2i-j-k and, i=7j -3k, , 2-1 -1=0, , (ii) d =2i-j-2k, 6 =i+2j-3k and, , 1-3+7)3 (6+) + 1(14 ), , 3-4+7k, , 4+-18-3 +14+ =0, , (Try yourself) ii), , 3, , 7, , 42 ii) 40., , a,b,are coplanar, where ), , =2i, , b=i+2+ 3k and =3i +aj +5k, vectors are, , 2 -1 1, , coplanar., , I2 3=0, , 3 a5, 2(10-3)+1 (5-9)+1(A-6)=0, Student's iuminatopr, , =0, , ()=i-j+k, b =3i +j + 2k and, vectors, , Given, , 3, , A=0, , Find the value of A for which the, , Ans:, , vectors are, , -j+k,, , =i+aj-3k, Ans: Given vectors, , are, , coplanar., , 1-1, 3, , 1, , 2=0, -3, , 1(-3-2)+1(-9-2) +1(3A -1) =0, -3-2-11+3-1=0, A=15., , I PU Mathematics