VECTOR ALGEBRA. Syllabus :

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1 MV VECTOR ALGEBRA Syllus : Vetors nd Slrs, ddition of vetors, omponent of vetor, omponents of vetor in two dimensions nd three dimensionl spe, slr nd vetor produts, slr nd vetor triple produt. Einstein Clsses, Unit No. 0, 0, Vrdhmn Ring Rod Plz, Viks Puri Extn., Outer Ring Rod New Delhi 0 08, Ph. : 96905, 857

2 MV CONCEPTS CA. Vetors & Their Representtion : Vetor quntities re speified y definite mgnitude nd definite diretions. A vetor is generlly represented y direted line segment, sy AB. A is lled the initil point & B is lled the terminl point. The mgnitude of vetor AB is epxressed y AB. CB. Types of Vetors Zero Vetor : A vetor of zero mgnitude is zero vetor i.e. whih hs the sme initil & terminl point, is lled Zero Vetor. It is denoted y O. The diretion of zero vetor is indeterminte. Unit Vetor : A vetor of unit mgnitude in the diretion of vetor is lled unit vetor long nd is denoted y â symolilly, â. Equl Vetors : Two vetors re sid to e equl if they hve the sme mgnitude, diretion & represent the sme physil quntity. Colliner Vetors : Two vetors re sid to e olliner if their direted line segment re prllel irrespetive of their diretions. Colliner vetors re lso lled prllel vetors. If they hve the sme diretion they re nmed s like vetors otherwise unlike vetors. Symolilly, two non zero vetors nd re olliner if nd only if, K, where K R Vetors î ĵ kˆ nd î ĵ kˆ re olliner if Coplnr Vetors : A given numer of vetors re lled oplnr if their line segments re ll prllel to the sme plne. Note tht Two Vetors Are Alwys Coplnr. C. Slr Produt Of Two Vetors : If nd re. = os.. i.i = j.j = k.k = ; i.j = j.k = k.i = 0 projetion of on. if = i + j + k & = i + j + k then. = + +, Vetor Produt Of Two Vetors :. If nd re two vetors nd is the ngle etween them then = sin nˆ, where nˆ is the unit vetor perpendiulr to oth nd suh tht, nd n forms right hnded srew system.. î î ĵ ĵ kˆ kˆ 0;î ĵ kˆ, ĵkˆ î,kˆ î ĵ. If î ĵ kˆ & î ĵ kˆ then î ĵ kˆ Einstein Clsses, Unit No. 0, 0, Vrdhmn Ring Rod Plz, Viks Puri Extn., Outer Ring Rod New Delhi 0 08, Ph. : 96905, 857

3 MV Prtie Prolems :. If nd re two vetors, suh tht. 0 nd., then the ngle etween vetors nd is 7/4 /4 /4. The vetors is perpendiulr to 7 5 nd 5 is perpendiulr to 7. The ngle etween nd is 6. If, 5 nd 8., then is equl to none of these 4. The vlues of so tht for ll rel x, the vetors xî 6ĵ kˆ nd xî ĵ xkˆ mke n otuse ngle with eh other, re none of these 5. If A (,,),C (0,, ) re two given vetors, then vetor B stisfying the eqution AB C nd A.B is 6. If,, 5 5,,,, re unit vetors suh tht 0 5,, then the vlue of... 5,, none of these 7. If nd re unit vetors nd is the ngle etween them, then the vlue of suh tht is unit vetor, is [Answers : () d () () (4) (5) d (6) (7) ] none of these C Slr Triple Produt : Slr triple produt geometrilly represents the volume of the prllelopiped whose three oterminous edges re represented y, & i.e. V [ ]. In slr triple produt the position of dot & ross n e interhnged i.e..( ) (). OR [ ] [ ] [ ].( ).() i.e. [ ] [ ] If i j k; i j k & i j k then [ ] Einstein Clsses, Unit No. 0, 0, Vrdhmn Ring Rod Plz, Viks Puri Extn., Outer Ring Rod New Delhi 0 08, Ph. : 96905, 857

4 MV 4 Prtie Prolems :. The volume of prllelopiped whose sides re given y OA i j,ob i j k,oc i k is 4/ 4 /7 None of these. If = i j + k, = i j k nd = i + pj + 5k re oplnr, then p = 6 6. If, nd re unit oplnr vetors, then the slr triple produt [ ] 0 4. Let î ĵ kˆ, î ĵ kˆ nd unit vetor e oplnr. If is perpendiulr to, then = ( ĵ kˆ ) ( î ĵ kˆ ) (î ĵ) (î ĵ kˆ ) 5 [Answers : () () () (4) ] C4 Vetor Triple Produt : Let,, e ny three vetors, then the expression ( ) is vetor & is lled vetor triple produt. ( ) (.) (.) () (.) (.) Einstein Clsses, Unit No. 0, 0, Vrdhmn Ring Rod Plz, Viks Puri Extn., Outer Ring Rod New Delhi 0 08, Ph. : 96905, 857

5 MV 5 INITIAL STEP EXERCISE. If is perpendiulr to +, is perpendiulr to +, is perpendiulr to +, =, = nd = 6, then + + = none of these. The points with position vetors 60i + j, 40i 8j nd i 5j re olliner if none of these. Four points 7i 4j + 7k, i 6j + 0k, i j + 4k nd 5i j + k form prllelogrm squre retngle rhomus 4. Unit vetor in xy-plne, tht mkes ngles of 45 0 nd 60 0 with i + j nd i 4j respetively is i (i + j)/ (i j)/ None 5. If 0, then is equl to 6( ) 6( ) 6( ) none of these 6. (.î) (.ĵ) (.kˆ ) is equl to.(i + j + k) none of these 7. If,, re three vetors of equl mgnitude nd the ngle etween eh pir of vetors is / suh tht + + = 6 then is equl to 6/ 8. Are of the prllogrm whose digonls represent the vetors i + j k nd i j + 4k is Let,, e three unit vetors nd. =. = 0. If the ngle etween nd is / then () is equl to / / none of these 0. If A( x,, ), B(, y, ), C(,, z) nd D(,, ) re oplnr, then x y z x + y + z = x y z none of these. Let = i + j nd = j + k. If = + where is olliner with nd is perpendiulr to, then = i + j (i + j k)/ (i j + 4k)/ none of these. Let,, e distint rel numers. The points with position vetors i + j + k, i + j + k, i + j + k re olliner form n equilterl tringle form n isoseles tringle form right ngled tringle. The point hving position vetors i + j + 4k, i + 4j + k, 4i + j + k re the verties of right ngled tringle isoseles tringle equilterl tringle olliner 4. If = i + j + k, = 4i + j + 4k nd = i + j + k re linerly dependent vetors nd =, then =, = =, = ± =, = ± = ±, = 5. If the slr produt of the vetor i + j + k with unit vetor prllel to the sum of vetors i + 4j 5k nd i + j + k is equl to one, then is equl to 5 6. If eh of pirs p, q nd r, s represented plne, then the plnes re perpendiulr to eh other if (p q). (r s) = 0 (p q) (r s) = 0 (p r). (q s) = 0 None of these Einstein Clsses, Unit No. 0, 0, Vrdhmn Ring Rod Plz, Viks Puri Extn., Outer Ring Rod New Delhi 0 08, Ph. : 96905, 857

6 MV 6 7. The numer of vetors of unit length perpendiulr to vetors = (,, 0) nd = (0,, ) is one two three infinite. A unit vetor whih is eqully inlined to the i j k 4j k vetors i, nd 5 is ( i + 5j 5k)/5 (i + 5j 5k)/5 (i + 5j + 5k)/5 (i 5j 5k)/5. A vetor of mgnitude 5 long isetor of the ngle etween the two vetors i j + k nd i + j k is 5 (i k) / 0 5(i 4j + k) / 6 5(i + 4j + k) / 6 none of these. Let,, e the three unit vetors suh tht ( ) = ( + ) / nd the ngles etween, nd, e nd respetively then = /4, = /4 = /4, = /4 = /4, = /4 none of these 4. Let p, q, r e three mutully perpendiulr vetors of the sme mgnitude. If vetor x stisfies the eqution : p {(x q) p} + q {(x r) q} + r {x p) r} = 0 then x is given y (p + q r) / r (p +q + r) / 5. (p + q + r) / (p + q + r) / If,, re the three non-oplnr vetors nd p,q, r re vetors defined y the reltions p,q, r [ ] [ ] [ ] then the vlue of the expression ( ).p ( ).q ( ).r is equl to 0 6. A, B, C nd D re four points in plne with position vetors,, nd d respetively suh tht ( d).( ) ( d).( ) 0. Then for the ABC, D is its 8. If,, re non-oplnr unit vetors suh tht ( ) = + / then the ngle etween nd is /4 /4 / FINAL STEP EXERCISE inentre irumentre orthoentre entroid 7. A, B, C re three vetors respetively given y i + k, i + j + k nd 4i j + 7k, then the vetor R whih stisfies the reltions R B = C B nd R. A = 0 is i 8j + k i + 4j + k i 8j + k None 8. A vetor hs omponents p nd with respet to retngulr drtesin system. This system is rotted through ertin ngle out the origin in the ounter-lokwise sense. If with respet to new system, hs omponents p + nd, then p = 0 p = or p = / p = or p = / p = or p = 9. Let = i j + k, = i + j k nd = i + j k e three vetors. A vetor in the plne of nd whose projetion on is of mgnitude (/) is i + j k i + j + k i j + 5k oth nd re orret 0. Let the vetors,, nd d e suh tht ( ) ( d) = 0. Let P nd P e plnes determined y the pirs of vetors, nd, d respetively. Then the ngle etween P nd P is 0 /4 / /. Let p nd q e the position vetors of P nd Q respetively, with respet to O nd p = p, q = q. The points R nd S divide PQ internlly nd externlly in the rtio : respetively. If OR nd OS re perpendiulr, then 9p = 4q 4p = 9q 9p = 4q 4p = 9q. Let = i + j k nd = i + j. If is vetor suh tht. =, = nd the ngle etween ( ) nd is 0 0, then ( ) = / / Einstein Clsses, Unit No. 0, 0, Vrdhmn Ring Rod Plz, Viks Puri Extn., Outer Ring Rod New Delhi 0 08, Ph. : 96905, 857

7 A N S W E R S (Initil Step Exerise) MV 7 A N S W E R S (Finl Step Exerise) d.. 9. d 4. d d 5. d d AIEEE ANALYSIS [00]. Given two vetors î ĵ nd î ĵ, the unit vetor oplnr with the two vetors nd perpendiulr to first is (î ĵ) (î ĵ) (î ĵ) none of these 5. If the vetors, xî yĵ zkˆ nd ˆ ĵ re suh tht form right hnded system then is zî xkˆ 0,nd 4. The vetor î xĵ kˆ is rotted through n ngle nd douled in mgnitude, then it eomes 4î (4x )ĵ kˆ. The vlue of x is (, 7),,,0 y ĵ zî xkˆ. If the vetors, nd from the sides BC, CA nd AB respetively, of tringle ABC, then Einstein Clsses, Unit No. 0, 0, Vrdhmn Ring Rod Plz, Viks Puri Extn., Outer Ring Rod New Delhi 0 08, Ph. : 96905, 857

8 MV 8 AIEEE ANALYSIS [00] 5. Consider point A, B, C nd D with position vetors 7î 4 ĵ 7kˆ, î 6 ĵ 0kˆ, î ĵ 4kˆ nd 5î ĵ 5kˆ respetively. Then ABCD is rhomus retngle squre none 6. If u, v nd w re three non-oplnr vetors, then (u v w).[(u v) (v w)] equls u.v w u.w v u.v w 0 7. Let u i j, v i j nd w i j k. If nˆ is unit vetor suh tht u.n 0 nd v.n 0, then w. n is equl to 0 8. The vetors AB î 4kˆ nd AC re the sides of tringle ABC. 5î ĵ 4kˆ The length of the medin through A is ,, 0,,,... is equl to 9. re vetors, suh tht If 0, then nd vetors (,, ), (,, ) nd (,, ) re non-oplnr, then the produt equls 0 AIEEE ANALYSIS [004/005]. Let, nd e three non-zero vetors suh tht no two of these re olliner. If the vetors is olliner with nd is olliner with ( eing some non-zero slr) then 6 equls [004] 0. A prtile is ted upon y onstnt fores 4î ĵ kˆ nd î ĵ kˆ whih disple it form point î ĵ kˆ to the point 5î 4ĵ kˆ. The work done in stndrd units y the fores is given y [004]. If,, re non-oplnr vetors nd is rel numer, then the vetors, 4 nd ( ) re non oplnr for ll exept two vlues of ll exept one vlue of ll vlues of no vlue of [004] 4. Let u, v, w e suh tht u, v, w. If the projetion v nd u is equl to tht of w long u nd v, w re perpendiulr to eh other then u v w equls [004] Einstein Clsses, Unit No. 0, 0, Vrdhmn Ring Rod Plz, Viks Puri Extn., Outer Ring Rod New Delhi 0 08, Ph. : 96905, 857

9 5. Let, nd e non-zero vetors suh tht ( ). If is the ute ngle etween the vetors nd then sin equls [004] 6. If C is the mid point of AB nd P is ny point outside AB, then PA PB PC PA PB PC PA PB PC 0 PA PB PC 0 [005] [Ans. : ] 7. For ny vetor, the vlue of ( î ) ( ĵ) ( kˆ ) [005] 4 is equl to MV 9 8. Let î kˆ, xî ĵ ( x)kˆ nd. yî xĵ ( x y)kˆ. Then,, depends on only x only y neither x nor y oth x nd y [005] 9. Let, nd e distint non-negtive numers. If the vetors î ĵ kˆ,î kˆ nd î ĵ kˆ lie in plne, then is the Arithmeti Men of nd the Geometri Men of nd the Hrmoni Men of nd equl to zero [005] 0. If,, re non-oplnr vetors nd is rel numer then ( ) for no vlue of extly one vlue of extly two vlues of extly three vlues of [005] AIEEE ANALYSIS [006]. If ( ) ( ), where, nd re ny three vetors suh tht. 0,. 0, then AD AB AC (AB) (AC) nd re prllel inlined t n ngle of / etween them inlined t n ngle of /6 etween them perpendiulr. ABC is tringle, right ngles t A. The resultnt of the fores ting long AB nd AC AB, AC with mgnitude respetively is the fore long AD, when D is the foot of the perpendiulr from A onto BC. The mgnitude of the resultnt is (AB)(AC) AB AC AB AC. The vlues of, for whih the points A, B, C with position vetors î ĵ kˆ, î ĵ 5kˆ nd î ĵ kˆ respetively re the verties of right-ngled tringle with C re nd nd nd nd Einstein Clsses, Unit No. 0, 0, Vrdhmn Ring Rod Plz, Viks Puri Extn., Outer Ring Rod New Delhi 0 08, Ph. : 96905, 857

10 MV 0 AIEEE ANALYSIS [007] 4. If û nd vˆ re unit vetors nd is the ute ngle etween them, then û vˆ is unit vetor for No vlue of Extly one vlue of Extly two vlues of More thn two vlues of 5. Let î ĵ kˆ, î ĵ kˆ nd xî (x )ĵ kˆ. If the vetor lies in the plne of nd, then x equls 4 0 ANSWERS AIEEE ANALYSIS d d d If r stisfies the eqution r (î ĵ kˆ ) î kˆ, then for ny slr t, r is equl to î t(î ĵ kˆ ) ĵ t(î ĵ kˆ ) kˆ t(î ĵ kˆ ) î kˆ t(î ĵ kˆ ). For non-oplnr vetors, nd, ( ) holds if nd only if The vlue of î ĵ kˆ is none of these TEST YOURSELF 4. The vlue of î ( î) ĵ( ĵ) kˆ (kˆ ) is 0 5. The vetor isets the ngle etween nd if ANSWERS Einstein Clsses, Unit No. 0, 0, Vrdhmn Ring Rod Plz, Viks Puri Extn., Outer Ring Rod New Delhi 0 08, Ph. : 96905, 857

m m m m m m m m P m P m ( ) m m P( ) ( ). The o-ordinte of the point P( ) dividing the line segment joining the two points ( ) nd ( ) eternll in the r

m m m m m m m m P m P m ( ) m m P( ) ( ). The o-ordinte of the point P( ) dividing the line segment joining the two points ( ) nd ( ) eternll in the r CO-ORDINTE GEOMETR II I Qudrnt Qudrnt (-.+) (++) X X - - - 0 - III IV Qudrnt - Qudrnt (--) - (+-) Region CRTESIN CO-ORDINTE SSTEM : Retngulr Co-ordinte Sstem : Let X' OX nd 'O e two mutull perpendiulr