CA Vectos & Thei Repesenttion : CB MV VECTORS Vecto quntities e specified y definite mgnitude nd definite diections A vecto is genelly epesented y diected line segment, sy AB A is clled the initil point & B is clled the teminl point The mgnitude of vecto AB is epxessed y AB Types of Vectos Zeo o Null Vecto : A vecto of zeo mgnitude is zeo vecto ie which hs the sme initil & teminl point, is clled Zeo Vecto It is denoted y O The diection of zeo vecto is indeteminte o ity Unit Vecto : A vecto of unit mgnitude in the diection of vecto is clled unit vecto long nd is denoted y â symoliclly, â Equl Vectos : Two vectos e sid to e equl if they hve the sme mgnitude, diection & epesent the sme physicl quntity Colline Vectos : Two vectos e sid to e colline if thei diected line segment e pllel iespective of thei diections Colline vectos e lso clled pllel vectos If they hve the sme diection they e nmed s like vectos othewise unlike vectos Symoliclly, two non zeo vectos nd e colline if nd only if, K, whee K R Vectos î ĵ kˆ nd î ĵ kˆ e colline if Copln Vectos : A given nume of vectos e clled copln if thei line segments e ll pllel to the sme plne Note tht Two Vectos Ae Alwys Copln CC Angle Between two Vectos It is the smlle ngle fomed when the initil points o the teminl points of the two vectos e ought togethe It should e noted tht 0 0 80 0 C Position Vecto Of A Point : Let O e fixed oigin, then the position vecto of point P is the vecto OP If nd e position vectos of two points A nd B, then AB = position vecto of B position vecto of A Pctice Polems : Find unit vecto in the diection of AB, whee A(,, ) nd B(4, 5, 6) e the given points Find vecto in the diection of the vecto (î ĵ) tht hs mgnitude 5 units If, e the vectos foming consecutive sides of egul hexgon ABCDEF, expess the vectos CD, DE, EF, FA, AC, AD, AE nd CE in tems of nd Einstein Clsses, Unit No 0, 0, Vdhmn Ring Rod Plz, Viks Pui Extn, Oute Ring Rod New Delhi 0 08, Ph : 96905, 857
MV [Answes : () î ĵ AE, CE ] kˆ () 5 î 0 5 ĵ () CD, AF, 0 C Diection Rtios nd Diection Cosines of Vecto Conside vecto î ĵ ckˆ Then, the numes,, c e clled the diection tios of Diection cosines of e given y l, m nd n c c c c If l, m, n e the diection cosines of vecto then we lwys hve (l + m + n ) = Pctice Polems : If A(,, ) nd B(,, ) e two given points in spce then find (i) the diection tios of AB nd (ii) the diection cosines of AB [Answes : () (i) (, 4, 4) (ii),, ] C4 Distnce Fomul Distnce etween the two points A() nd B() Pctice Polems : is AB = Let,, e distinct el numes Pove tht the points with position vectos î ĵ kˆ, î ĵ kˆ, î ĵ kˆ fom n equiltel tingle C5 Section Fomul If nd e the position vectos of two points A & B then the postion vecto of point which divides AB n m in the tio m : n is given y : m n Note postion vecto of mid point of AB Pctice Polems : The position vectos of two vetices nd the centoid of tingle e î ĵ,î ĵ kˆ nd kˆ espectively Find the position vecto of the thid vetex of the tingle? [Answes : () î kˆ ] C6 Opetion on Vectos : C6A Addition Of Vectos : If two vectos & e epesented y OA & OB, then thei sum is vecto epesented y OC, whee OC is the digonl of the pllelogm OACB Einstein Clsses, Unit No 0, 0, Vdhmn Ring Rod Plz, Viks Pui Extn, Oute Ring Rod New Delhi 0 08, Ph : 96905, 857
(commuttive) ( ) ( (ssocitivity) 0 0 ( ) 0 ( ) MV A vecto in the diection of the isecto of the ngle etween the two vectos & is Hence isecto of the ngle etween the two vectos nd is (â ˆ ), whee R + Bisecto of the exteio ngle etween & is (â ˆ ), R + C6B Multipliction Of A Vecto By A Scl : If is vecto & m is scl, then multipliction is clled Scl Multipliction If m() ()m m (m n) m n C6C Scl Poduct Of Two Vectos : m is vecto pllel to whose modulus is m times tht of This m(n) n(m) (mn) m( ) m m â ndˆ e vectos & m, n scls, then : Geometicl intepettion of Scl Poduct Let nd e vecto epesented y OA nd OB espectively Let e the ngle etween OA nd OB Dw BL OA nd AM OB Fom s OBL nd OAM, we hve OL = OB cos nd OM = OA cos Hee OL nd OM e known s pojections of on nd on espectively Now, = cos Agin = (OB cos) = (OL) = (Mgnitude of ) (Pojection of on ) (i) = cos ( cos) = (OA cos) = (OM) = (mgnitude of ) (Pojection of nd Thus geometiclly intepeted, the scl poduct of two vectos is the poduct of modulus of eithe vecto nd the pojection of the othe in its diection ii = jj = kk = ; ij = jk = ki = 0 pojection of on (ii) Einstein Clsses, Unit No 0, 0, Vdhmn Ring Rod Plz, Viks Pui Extn, Oute Ring Rod New Delhi 0 08, Ph : 96905, 857
if = i + j + k & = i + j + k then = + +, the ngle etween nd is given y cos 0 4 cos (0 ), note tht if is cute then > 0 if is otuse then < 0 5, (commuttive) ( c (distiutive) 6 0 ( 0, 0) 7 m m m (ssocitive) whee m is scl Note : (i) Mximum vlue of is (ii) Minimum vlue of is (iii) Any vecto cn e witten s, (î)î (ĵ)ĵ (kˆ ) kˆ Pctice Polems : Find the pojection of î ĵ kˆ on î ĵ kˆ MV 4 If,, c e thee vectos such tht c 0 nd, 5, c 7, find the ngle etween nd If â nd ˆ e unit vectos nd is the ngle etween them, pove tht sin â ˆ 4 If,, c e unit vectos such tht c 0 then find the vlue of ( c 5 Fo ny vecto in spce, show tht ( î)î (ĵ)ĵ (kˆ )kˆ 6 If (î ĵ kˆ ), ( î ĵ kˆ ) nd c (î ĵ) such tht ( ) c, then find the vlue of 7 If A(0,, ), B(,, 5) nd C(0,, ) e the vetices of ABC, show, using vectos, tht ABC is ight ngled t C 8 Show tht the points A(,, ), B(,, 5) nd C(, 4, 4) e the vetices of ight-ngled tingle Also, find the emining ngles of the tingle 9 Let ( î ĵ kˆ ),(î 5ĵ),(î ĵ kˆ ) nd (î 6ĵ kˆ ) e the position vectos of points A, B, C, D espectively Find the ngle etween AB nd CD Hence, show tht AB CD 0 Expess the vecto (5î ĵ 5kˆ ) s sum of two vectos such tht one is pllel to the vecto (î kˆ ) nd the othe is pependicul to The dot poducts of vecto with the vectos ( î ĵ kˆ ), (î ĵ kˆ ) nd (î ĵ 4kˆ ) e 0, 5, 8 espectively Find the vecto Find the pojection of î ĵ kˆ in the diection of vecto î ĵ kˆ detemined y the pojection Wht is the vecto Einstein Clsses, Unit No 0, 0, Vdhmn Ring Rod Plz, Viks Pui Extn, Oute Ring Rod New Delhi 0 08, Ph : 96905, 857
If 4î 6ĵ nd ĵ 4kˆ, find the vecto fom of the component of nd MV 5 4 Let u, v, w e vectos such tht u v w 0 If u, v 4, w 5, then pove tht the vlue of uv vu w u is 5 [Answes : () 5 6 () 60 0 (4) / (6) 8 (8) A cos 5 4 nd B cos (0) ( 6î kˆ ) nd ( î ĵ kˆ ) () î ĵ kˆ () (î ĵ kˆ ) 8 8 () OP ˆ (ĵ 4kˆ ) ] 7 5 5 6 4 C6D Vecto Poduct Of Two Vectos : If nd e two vectos & is the ngle etween them then = sin n, whee n is the unit vecto pependicul to oth nd such tht,& n foms ight hnded scew system Geometiclly = e of the pllelogm whose two djcent sides e epesented y nd î î ĵ ĵ kˆ kˆ 0;î ĵ kˆ, ĵkˆ î,kˆ î ĵ 4 If î ĵ kˆ & î ĵ kˆ then î ĵ kˆ 5 (not commuttive) 6 m m m (ssocitive) whee m is scl 7 ( () ( (distiutive) 8 0 e pllel (colline) ( 0, 0) ie 9 Unit vecto pependicul to the plne of nd is nˆ K, whee K is scl A vecto of mgnitude & pependicul to the plne of If is the ngle etween & then sin If, & c nd is e the pv s of points A, B & C then the vecto e of tingle ABC = c c The points A, B & C e colline if c c 0 Ae of ny qudiltel whose digonl vectos e d & d is given y d d Einstein Clsses, Unit No 0, 0, Vdhmn Ring Rod Plz, Viks Pui Extn, Oute Ring Rod New Delhi 0 08, Ph : 96905, 857
MV 6 Lgnge s Identity : fo ny two vectos nd ;( ) () Pctice Polems : Find the e of the pllelogm whose digonls e epesented y the vectos (î ĵ kˆ ) nd d (î 4ĵ kˆ ) d Find the e of the tingle whose djcent sides e detemined y the vectos ( î 5kˆ ) nd (î ĵ kˆ ) Show tht the points whose position vectos e ( 5î 6ĵ 7kˆ ), (7î 8ĵ 9kˆ ) nd (î 0ĵ 5kˆ ) e colline 4 Show tht the points hving position vectos (, (, ( 8 ) e colline, whteve e,, c 5 Pove tht 6 Pove tht ( (c ) c( ) 0 7 If c 0, pove tht ( ) ( ( 8 Pove tht the points A, B, C with position vectos,, c e colline if nd only if ( ( () 0 9 If cd nd c d, show tht ( d) ( c ), it eing given tht d nd c 0 If î ĵ kˆ nd ĵ kˆ, find vecto c such tht c nd c [Answes : () 55 sq units () 65 sq units ] C6E Scl Tiple Poduct : The scl tiple poduct of thee vectos, & c is defined s : c c sin cos whee is the ngle etween & & is the ngle etween & c It is lso witten s [ c ] nd spelled s ox poduct Scl tiple poduct geometiclly epesents the volume of the pllelopiped whose thee coteminous edges e epesented y, & c ie V [ c] In scl tiple poduct the position of dot & coss cn e intechnged ie ( ()c OR [ c] [ c ] [c ] ( ( ie [ c] [ c ] If i j k; i j k & c c i c j c k then [ c] c c c Einstein Clsses, Unit No 0, 0, Vdhmn Ring Rod Plz, Viks Pui Extn, Oute Ring Rod New Delhi 0 08, Ph : 96905, 857
MV 7 In genel, if l m n; l m n & c c l c m c n e non copln vectos then c [l mn]; whee l,m & n c c c If,, c e copln [ c] = 0 Scl poduct of thee vectos, two of which e equl o pllel is 0 ie [ c] 0 If,, c e non-copln then [ c] 0 fo ight hnded system & [ c] 0 fo left hnded system [ i jk] K[ c] K[ c] K[ c] [( )cd] [cd] [ cd] The volume of the tethedon OABC with O s oigin & the pv s of A, B nd C eing, & c espectively is given y V [ c] 6 The position vecto of the centoid of tethedon in the pv s of its vetices e,,c & d e given y [ c d] 4 Note tht this is lso the point of concuency of the lines joining the vetices to the centoids of the opposite fces nd is lso clled the cente of the tethedon In cse the tethedon is egul it is equidistnt fom the vetices nd the fou fces of the tethedon Rememe tht : [ c c ] 0 & [ c c ] [ c] Pctice Polems : Find the volume of the pllelopiped whose edges e epesented y the vectos () (,,4), (,, ), c (,,) () (,,0), (,, ), c (,0, ) The volume of the pllelopiped whose edges e epesented y î kˆ, ĵ kˆ, î ĵ 5kˆ is 540 Find the vlue of If the vectos î ĵ kˆ, î ĵ kˆ nd î ĵ ckˆ (, c ) e copln, then pove tht the vlue of is c 4 If,, c e thee vectos, pove tht (i) [ c c ] [ c] (ii) [ c c ] 0 [ c c] c (iii) 5 Find the volume of pllepiped whose sides e given y : i j 4k, i j k nd i j k 6 Find so tht the following vectos e copln i 4j 5k, i j k, i j 5k Einstein Clsses, Unit No 0, 0, Vdhmn Ring Rod Plz, Viks Pui Extn, Oute Ring Rod New Delhi 0 08, Ph : 96905, 857
MV 8 7 Find the volume of pllelopiped whose sides e given y i 7j 5k, 5i 7j k nd 7i 5j k 8 Pove tht : c c c 9 Pove tht : c c 0 0 Find the constnt so tht the vecto î ĵ kˆ, î ĵ kˆ, c î ĵ 5kˆ e copln If xa 0, xb 0, xc 0 fo some non-zeo vecto x, then pove tht A B C = 0 [Answes : (5) 7 (6) 6/5 (7) 64] C6F Vecto Tiple Poduct : Let,, c e ny thee vectos, then the expession ( is vecto & is clled vecto tiple poduct Geometicl Intepettion of ( Conside the expession ( which itself is vecto, since it is coss poduct of two vectos &( Now ( is vecto pependicul to the plne contining &( ut ( is vecto pependicul to the plne contining & c, theefoe ( is vecto which lies in the plne of c nd pependicul to Hence we cn expess ( in tems of & c ie ( = x yc whee x & y e scls ( ( () c () c ( ( ( ) c (, in genel Pctice Polems : Pove tht ( (c ) c ( ) 0 If î ĵ kˆ, î ĵ kˆ nd c î ĵ kˆ, veify tht ( ( ( ) c Fo ny non-zeo vecto, show tht î ( î) ĵ( j) kˆ ( k) 4 If,, c e thee vectos, pove tht ( ) c ( if nd only if ( c ) 0 5 Given tht î ĵ kˆ,ˆ î ĵ 7kˆ nd c 5î 6ĵ 5kˆ, veify tht ( ( () c C7 Recipocl System Of Vectos : If,,c &,, c e two sets of non copln vectos such tht cc systems e clled Recipocl System of vectos c c Note : c [ c] [ c] [ c] then the two Einstein Clsses, Unit No 0, 0, Vdhmn Ring Rod Plz, Viks Pui Extn, Oute Ring Rod New Delhi 0 08, Ph : 96905, 857
MV 9 C8 Line Comintions : Given finite set of vectos,, c then the vecto x y zc is clled line comintion of,, c fo ny x, y, z, R We hve the following esults : () If, e non zeo, non-colline vectos then x y x y x x ;y y () Fundmentl Theoem : Let, e non-zeo, non colline vectos Then ny vecto copln with, cn e expessed uniquelly s line comintions of, ( If,, c e non-zeo, non-copln vectos then : x y zc x y zc x x,y y, z z (d) Fundmentl Theoem In spce : Let,, c e non-zeo, non-copln vectos in spce Then ny vecto, cn e uniquly expessed s line comintion of,, c ie Thee exist some unique x, y R such tht x y zc (e) If x,x xn e n non zeo vectos, & k, k, k n e n scls & if the line comintion kx k x k nxn 0 k 0,k 0k n 0 then we sy tht vectos x,x e LINEARLY INDEPENDENT VECTORS xn (f) If x,x xn e not LINEARLY INDEPENDENT then they e sid to e LINEARLY INDEPENDENT VECTORS ie if kx k x k nxn 0 & if thee tlest one k 0 then x,x e sid to e xn Linely Dependent : Note : If k 0;k x k x k x k x k x 0 Note : ie k x k x x x k k x x k k x c x c x k k x k c k x x k c x k k n n k x x c x is expessed s line comintion of vectos,x,x, x, xn Hence x vectos with x,x,x,x x n n n k n n k n foms linely dependent set of If î ĵ 5kˆ then is expessed s Line Comintion of vectos î, ĵ, kˆ fom linely dependent set of vectos In genel, evey set of fou vectos is linely dependent system î, ĵ, kˆ e Linely Independent set of vectos Fo x n Kî K ĵ K kˆ 0 K K K 0 Two vectos & e linely dependent is pllel to ie 0 & Convesely if & then & e linely independent line dependence of Einstein Clsses, Unit No 0, 0, Vdhmn Ring Rod Plz, Viks Pui Extn, Oute Ring Rod New Delhi 0 08, Ph : 96905, 857
MV 0 Note : Test Of Collineity : Thee points A, B, C with position vectos,, c espectively e colline, if & only if thee exist scls x, y, z not ll zeo simultneously such tht; x y z c 0, whee x + y + z = 0 Note : Test Of Coplnity : Fou points A, B, C, D with position vectos,, c, d espectively e copln if nd only if thee exist scls x, y, z, w not ll zeo simultneously such tht x y z c wd 0 whee, x + y + z + w = 0 Pctice Polems : Let î ĵ kˆ, 4î ĵ 4kˆ nd c, then find the vlue of nd c î ĵ kˆ e linely dependent vectos nd Einstein Clsses, Unit No 0, 0, Vdhmn Ring Rod Plz, Viks Pui Extn, Oute Ring Rod New Delhi 0 08, Ph : 96905, 857
MV SINGLE CORRECT CHOICE TYPE If is pependicul to c, is pependicul to c, c is pependicul to, =, = nd c = 6, then c = () 4 () 9 ( 7 (d) Let = i + j k nd = i + j If c is vecto such tht c = c, c = nd the ngle etween ( ) nd c is 0 0, then ( ) c = () / () / ( (d) A unit vecto which is eqully inclined to the i j k 4j k vectos i, nd is 5 () () ( (d) ( i + 5j 5k)/5 (i + 5j 5k)/5 (i + 5j + 5k)/5 (i 5j 5k)/5 4 Resolved pt of the vecto long the vecto is nd tht pependicul to is µ Then () = () () = () () () ( µ ( ) (d) µ 5 A vecto of mgnitude 5 long isecto of the ngle etween the two vectos i j + k nd i + j k is () () ( (d) 5 (i k) / 0 5(i 4j + k) / 6 5(i + 4j + k) / 6 none of these 6 Let â,ˆ, ĉ e the thee unit vectos such tht â (ˆ ĉ) = ( ˆ ĉ) / nd the ngles etween â, ĉ nd â, ˆ e nd espectively then () = /4, = /4 () ( = /4, = /4 = /4, = /4 (d) none of these 7 Let u i j, v i j nd w i j k If n is unit vecto such tht un 0 nd vn 0, then w n is equl to () () ( (d) 0 8 The vectos AB î 4kˆ nd AC e the sides of tingle ABC 5î ĵ 4kˆ The length of the medin though A is () 7 () ( 88 (d) 8 9 Let p,q, e thee mutully pependicul vectos of the sme mgnitude If vecto x stisfies the eqution : p{(x q) p} q{(x ) q)} {(x p) } 0 then x is given y () (p q ) () (p q ) / ( (p q ) / (d) (p q )/ 0 If,, c e the thee non-copln vectos nd p,q, e vectos defined y the eltions c c p,q, [ c] [ c] [ c] then the vlue of the expession ( )p ( q (c ) is equl to () 0 () ( (d) Einstein Clsses, Unit No 0, 0, Vdhmn Ring Rod Plz, Viks Pui Extn, Oute Ring Rod New Delhi 0 08, Ph : 96905, 857
MV If the non-zeo vectos nd e pependicul to ech othe, then the solution of the eqution is () x ( ) () x ( ) ( (d) x x A, B, C nd D e fou points in plne with position vectos,, c nd d espectively such tht ( d)( ( d)(c ) 0 Then the point D of the tingle ABC is () incente () cicumcente ( othocente (d) centoid A,B, C e thee vectos espectively given y i + k, i + j + k nd 4i j + 7k, then the vecto R which stisfies the eltions R B C B nd R A 0 is () i 8j + k () i + 4j + k ( i 8j + k (d) None 4 A vecto hs components p nd with espect to ectngul dtesin system This system is otted though cetin ngle out the oigin in the counte-clockwise sense If with espect to new system, hs components p + nd, then () p = 0 () p = o p = / ( p = o p = / (d) p = o p = 6 Let the vectos,, c nd d e such tht ( ) (c d) = 0 Let P nd P e plnes detemined y the pis of vectos, nd c, d espectively Then the ngle etween P nd P is () 0 () /4 ( / (d) / 7 Let p nd q e the position vectos of P nd Q espectively, with espect to O nd p = p, q = q The points R nd S divide PQ intenlly nd extenlly in the tio : espectively If e pependicul, then OR nd OS () 9p = 4q () 4p = 9q ( 9p = 4q (d) 4p = 9q 8 If u = i ( i) + j ( j) + k ( k), then () u is unit vecto () u = + i + j + k ( u = (d) u = 8 (i + j + k) 9 If u, v nd w e thee non-copln vectos, then (u v w)(u v) (v w) equls () uv w () uw v ( uv w (d) 0 0 Let,, c e distinct non-negtive numes If the vectos i + j + ck, i + k nd ci + cj + k lie in plne, then c is () () ( (d) the Aithmetic Men of nd the Geometic Men of nd the Hmonic Men of nd equl to zeo 5 If the vectos, nd c fom the sides BC, CA nd AB espectively, of tingle ABC, then () + c + c = 0 () ( = c = c = c = c (d) + c + c = 0 Einstein Clsses, Unit No 0, 0, Vdhmn Ring Rod Plz, Viks Pui Extn, Oute Ring Rod New Delhi 0 08, Ph : 96905, 857
MV EXCERCISE BASED ON NEW PATTERN COMPREHENSION TYPE Compehension- The vecto eqution of plne is eltion stisfied y position vectos of ll the points on the plne If P is plne nd nˆ is unit vecto though oigin which is pependicul to the plne P then vecto eqution of the plne must e nˆ d whee d epesents pependicul distnce of plne P fom oigin If A is point with position vecto then pependicul distnce of A fom the plne nˆ d must e () d nˆ () d nˆ ( d (d) None of these If e the foot of the pependicul fom A to the plne nˆ d then must e () ( (d nˆ ) nˆ () (d nˆ ) nˆ nˆ (d) none of these The position vecto of the imge of the point in the plne nˆ d must e (d 0) () nˆ () (d nˆ ) nˆ ( (d nˆ ) nˆ (d) None of these Compehension- Let ABC e tingle, AD, BE nd CF e the ngul isectos of its inteio ngles These isectos e concuent t point I clled incente of the tingle We know fom geomety tht BD AB DC AC If BC =, CA = nd AB = nd with efeence to some oigin let,, c e position vectos of A, B nd C espectively Then 4 The position vecto of I must e c () () c ( (d) c None of these 6 If is pependicul distnce of I fom the side BC then () () ( (d) IB IC must e cosec cosec B cosec None of these Compehension- Let xˆ, ŷ nd ẑ C cosec B C cosec sin A B C cosec sin A e unit vectos such tht xˆ ŷ ẑ,xˆ (ŷ ẑ), ( xˆ ŷ) ẑ c, xˆ 7 ŷ nd 4 7 The vlue of xˆ is (), 4 8 8 c () 4c c ( 4 (c ) (d) none 8 The vlue of ŷ is () 4 8 8 c () 4c c ( 4 (c ) (d) none 9 The vlue of ẑ is () 4 8 8 c () 4c c ( 4 (c ) (d) none 5 The vlue of IA IB IC must e () zeo () one ( + + (d) None of these Einstein Clsses, Unit No 0, 0, Vdhmn Ring Rod Plz, Viks Pui Extn, Oute Ring Rod New Delhi 0 08, Ph : 96905, 857
Compehension-4 A, B, C nd D e fou points such tht AB m(i 6 j k),bc ( i j) nd CD n( 6i 5 j k) Suppose two scls m nd n e such tht CD intesects AB t some point E 0 EB p AB,CE qcd, then which of the following is most ppopite? () 0 < p <, q > () 0 < p, q < ( p >, q > (d) 0 < q <, p > Vlue of m nd n fo which point of intesection E exists e () m > 0, n > 0 () ( Ae of tingle BCE is m,n m,n (d) m,n 0 () () 6 ( 6 (d) 6/ MATRIX-MATCH TYPE Mtching- Column - A Column - B (A) [ c c ] is (P) 0 equl to (B) If,, c e ny thee (Q) [ c] vectos in spce then (c ) (c )(c ) is equl to (C) If,, c e thee (R) [ c] noncopln vectos then [ c c ] is equl to (D) [ c c] is (S) [ c] equl to Mtching- MV 4 Column - A Column - B (A) If () ( (P) /6 nd 0 then [ c] is equl to (B) The cosine of the ngle (Q) 6/4 etween two digonls of cue is (C) The pojection of the (R) 0 vecto î ĵ kˆ on the line whose vecto eqution is ( t)î (t )j tk, eing the scl pmete, Einstein Clsses, Unit No 0, 0, Vdhmn Ring Rod Plz, Viks Pui Extn, Oute Ring Rod New Delhi 0 08, Ph : 96905, 857 is (D) If the vetices of (S) / tethedon hve the position vectos 0, i j,j k nd i k then the volume of the tethedon is Mtching- (T) None Column - A Column - B (A) Let u i j, v i j, (P) O (B) w i j k, If n is unit vecto such tht u n 0, v n 0, then the vlue of w n is î, ĵ, kˆ e mutully (Q) n oily pependicul unit vecto if j, then is (C) [ ( (c ()] (R) 8 is equl to (D) If d x( ) y( (S) ) z(c ) nd [c] = /8 nd x + y + z = k ( d then k = scle
MV 5 Mtching-4 Column - A Column - B (A) Let,, c e thee vectos(p) 5 nd c 0 If the ngle etween nd c is / then c (B) If c (C) is equl to = then (Q) / c c c c c c is equl to,, c e noncopln (R) ± vectos nd p,q, e defined s c c p,q, [ c] [c] [ c] ( ) p ( q (c ) is equl to (D) If,, c e thee (S) noncopln vectos epesented y concuent edges of pllelepiped of volume 4 then ( ) ( ( ( (c )( ) is equl to Mtching-5 Column - A Column - B (A) If thee unit vectos (P) /,,c stisfy c 0, then ngle etween nd is (B) If nd e two non- (Q) / colline unit vectos such tht (), then the ngle etween nd is equl to (C) In the pllelogm (R) infomtion ABCD if the intevl e isectos of the ngle B nd C intesect t the point P, then BPC is equl to (D) Let (S) / Then the ngle etween nd is inconsistent MULTIPLE CORRECT CHOICE TYPE The vectos (î ĵ kˆ ) is () () unit vecto mkes n ngled with the vecto ( î 4ĵ kˆ ) ( pllel to the vecto î ĵ kˆ (d) pependicul to the vectos î ĵ kˆ Which of the following expessions e meningful? () u (v w) () (uv) w ( (uv) w (d) u (vw) Let nd e two non-colline unit vectos If u () nd v then v is () u () u u ( u u (d) u u( ) 4 A vecto of mgnitude long isecto of the ngle etween the two vectos i j k nd i j k is () (i k) 0 () (i 4j k) 6 ( (i 4j k) 6 (d) none of these Einstein Clsses, Unit No 0, 0, Vdhmn Ring Rod Plz, Viks Pui Extn, Oute Ring Rod New Delhi 0 08, Ph : 96905, 857
MV 6 5 A unit vecto which is eqully inclined to the i j k 4j k vectos i, nd 5 is () ( i 5j 5k) 5 () (i 5j 5k) 5 ( (i 5j 5k) 5 (d) (i 5j 5k) 5 6 Thee points whose position vectos e,, c will e colline if () µ ( µ) c () c c 0 ( [ c] 0 (d) none of these 7 If,, c e noncopln nonzeo vectos nd is ny vecto in spce then [ c ] [c ] [ ] is equl to () [ c] () [ c] ( [ c ] (d) none of these 8 Let,, c e noncopln vectos nd c c p,q, then [ c] [ c ] [c ] () p () p q c ( p q c 0 (d) none of these 9 If x c nd x is equl to () ( ( () c ( (c ) (d) none of these 0 If,, c, d e ny fou vectos then () ( cd) is vecto () pependicul to,, c, d () long the line of intesection of two plnes, one contining, ( eqully inclined to oth nd cd (d) none of these Assetion-Reson Type Ech question contins STATEMENT- (Assetion) nd STATEMENT- (Reson) Ech question hs 4 choices (A), (B), (C) nd (D) out of which ONLY ONE is coect (A) (B) (C) (D) Sttement- is Tue, Sttement- is Tue; Sttement- is coect explntion fo Sttement- Sttement- is Tue, Sttement- is Tue; Sttement- is NOT coect explntion fo Sttement- Sttement- is Tue, Sttement- is Flse Sttement- is Flse, Sttement- is Tue STATEMENT- : Let (,µ (c ) nd c( ), µ, e copln STATEMENT- : Thee is line eltion etween them STATEMENT- : If the vectos nd e noncolline then the lines 6 c (c )nd c µ( ecopln STATEMENT- : Thee exist nd µ such tht the two vlues of ecome sme STATEMENT- : If thee points P, Q, R hve position vectos,, c espectively nd 5c 0 then the points P, Q, R must e colline STATEMENT- : If fo thee points A, B, C AB AC then the points A, B, C must e colline 4 STATEMENT- : If O is the cicumcente of the tingle ABC then OA OB OC 0 Einstein Clsses, Unit No 0, 0, Vdhmn Ring Rod Plz, Viks Pui Extn, Oute Ring Rod New Delhi 0 08, Ph : 96905, 857
MV 7 STATEMENT- : The position vecto of the cicumcente of tingle is c whee,, c e position vecto of vetices A, B, C espectively 5 STATEMENT- : Let,, c e thee unit vectos such tht 5 c 0 Then ( 0 STATEMENT- :,, c e copln vecto 6 STATEMENT- : Any fou o moe thn fou non-zeo vectos e lwys linely independent STATEMENT- : Two vectos e sid to e colline if they e eithe pllel o coincident 7 STATEMENT- : If und v e unit vectos inclined t n ngle nd x is unit vecto u v isecting the ngle etween then then x cos STATEMENT- : If ABC e n isosceles tingle with AB = AC = then vecto epesenting 8 STATEMENT- : If thee points P, Q, R hve position vectos,, c espectively nd 5c 0 then the points P, Q, R must e colline STATEMENT- : If fo thee points A, B, C AB AC then the points A, B, C must e colline 9 STATEMENT- : If vectos,ndc e copln then vectos, c nd c e lso copln STATEMENT- : [ c c] [ c] 0 STATEMENT- : If p, q e two noncolline nd nonzeo vectos such tht ( p q (c )p ( )q 0, whee,, c e the lengths of the sides of tingle, then the tingle is equiltel STATEMENT- : p,q,p q e noncopln isecto ngle A is given y AB AC AD (Answes) EXCERCISE BASED ON NEW PATTERN COMPREHENSION TYPE c 4 c 5 6 c 7 8 9 c 0 d MATRIX-MATCH TYPE [A-P ; B-R ; C-Q ; D-P] [A-R ; B-S ; C-Q ; D-P] [A-S ; B-Q ; C-P ; D-R] 4 [A-Q ; B-S ; C-S ; D-R] 5 [A-S ; B-R ; C-Q ; D-P] MULTIPLE CORRECT CHOICE TYPE,, c, c, c 4, c 5, d 6, 7, c 8, 9, c 0, c ASSERTION-REASON TYPE A A A 4 C 5 A 6 D 7 A 8 A 9 C 0 A Einstein Clsses, Unit No 0, 0, Vdhmn Ring Rod Plz, Viks Pui Extn, Oute Ring Rod New Delhi 0 08, Ph : 96905, 857
MV 8 INITIAL STEP EXERCISE (SUBJECTIVE) Show tht the digonls of pllelogm isect ech othe The sides of pllelogm e î 4ĵ 5kˆ nd î ĵ kˆ Find the unit vecto pllel to the digonl Pove tht the medins of tingle e concuent nd the point of concuency divides the medins in the tio : 4 In OAB, E is the mid-point of OB nd D is point on AB such tht AD : DB = : If OD nd AE intesect t P, detemine the tio OP : PD using vecto methods 5 The vecto î ĵ kˆ isects the ngle etween the vectos c nd long c 7î 4ĵ 4 6 If kˆ î 4ĵ Detemine the unit vecto nd î ĵ kˆ, detemine the vecto c of modulus 56 which is long the intenl isecto of the ngle etween the vectos nd 7 The position vectos of the points A nd B wt the oigin e î ĵ kˆ nd î ĵ kˆ espectively Detemine the vecto OP which isects the ngle AOB whee P is point on AB 8 The vecto xî yĵ zkˆ mkes with the plne of the two vectos,, nd,, n cute ngle cot () then show tht x(y + z) = yz 9 Given A = (,, 5), B = (5, 7, 9) nd C = (,, ) Find unit vecto noml to the plne of ABC 0 If nd lie on plne noml to the plne contining c nd d, show tht ( )(c d) 0 If the fou vectos,, c, d e copln, show tht ( ) (c d) 0 The vecto B ĵ 4kˆ is to e witten s the sum of vecto B, pllel to A î ĵ nd vecto B pependicul to A Then pove tht B (î ĵ) Find vecto of mgnitude 5 which mkes equl ngles with the vectos (î ĵ kˆ), ( 4î kˆ)nd c j 5 4 The vecto OA î ĵ kˆ tuns though ight ngle pssing though the positive x-xis on the wy Detemine the vecto in its new position 5 Thee vectos,, c e such tht they e of equl mgnitude nd ngle etween ech pi is sme If î ĵ nd ĵ kˆ, then detemine the vecto c 6 Find the pependicul distnce of A(, 4, ) fom the line BC whee the coodintes of B nd C e (,, ) nd (0, 5, ) espectively 7 Pove tht î ĵ 8 If,, c kˆ n n nc 0 e non-copln vectos nd Show tht n is null vecto 9 If,, c e copln vectos, pove tht c c c = 0 0 Pove tht (î )( ĵ) (î)(ĵ) Show tht (i) (î)î (ĵ) ĵ (kˆ) kˆ (ii) î ( î) ĵ ( j) k ( k) Einstein Clsses, Unit No 0, 0, Vdhmn Ring Rod Plz, Viks Pui Extn, Oute Ring Rod New Delhi 0 08, Ph : 96905, 857
Let OA,OB 0 ndoc whee O, A nd C e non-colline points Let p denotes the e of qudiltel OABC, nd let q denote the e of the pllelogm with OA nd OC s djcent sides If p = kq, then find the vlue of k? Vectos AB î ĵ kˆ nd CD î ĵ 4kˆ e not copln The position vectos of points A nd C e 6î 7ĵ 4kˆ nd 9 ĵ kˆ espectively Find the position vectos of point P on the line AB nd point Q on the line CD such tht PQ is pependicul to AB nd CD oth 4 (),, c e non-zeo vectos which e such tht c, c nd c then pove tht the thee vectos fom ight hnded othonoml tid () If c, c then pove tht,,c e mutully pependicul Also is unit vecto nd c nd e vectos of equl mgnitude 5 () Find vecto of mgnitude units copln with vectos î ĵ kˆ nd () î ĵ kˆ nd pependicul to vecto î ĵ kˆ Find the unit vecto which is copln with î ĵ kˆ nd î ĵ kˆ nd pependicul to î ĵ kˆ 6 An c AC of cicle sutends ight ngle t the cente O The point B divides the c in the tio : If OA nd OB, then clculte OC in tems of nd 7 If,, c e vectos such tht c, pove tht [( ) ( ( ( ] 0 MV 9 8 The mgnitude of vectos,, c e espectively, nd If ( 0, then find the cute ngle etween nd c? 9 Show tht ( ) c ( if nd only if nd c e colline o ( 0 0 Pove tht the points with position vectos, nd k e colline fo ll el vlues of k If,, c e unit vectos which stisfy the following conditions 0, c 0 nd ngle etween nd c is /6, then show tht ( Find the vecto which is pependicul to î ĵ 5kˆ nd î ĵ kˆ nd (î ĵ kˆ) 8 0 Two vetices of tingle e î ĵ nd î 5ĵ nd its othocente is t vecto of the thid vetex î ĵ Find the position 4 If the sum of two unit vectos e unit vecto the ngle etween them is / Also pove tht the modulus of thei diffeence is 5 If e nde e non colline unit vectos, compute e 5e )(e e ), if e e ( 6 A vecto A hs components A, A, A in ight hnded ectngul cetesin coodinte system O x, O y, O z The coodinte system is otted out the z-xis though n ngle Find the components of A in new coodinte system in tems of A, A, A 7 The xes of coodintes e otted out z-xis though n ngle of /4 in nti-clockwise diection nd components of vecto ecome, ; 4 Then show tht the components of the vecto in the old coodinte system e, 5, 4 8 If,, c e unit vectos, then find the mximum vlue of c c Einstein Clsses, Unit No 0, 0, Vdhmn Ring Rod Plz, Viks Pui Extn, Oute Ring Rod New Delhi 0 08, Ph : 96905, 857
9 nd e two unit vectos inclined t ny ngle to ech othe, then if 40 Pove tht ( (c ) c ( ) 0 4 If c e thee unit vectos such tht ( Find the ngles with mkes with nd c, nd c eing non-pllel MV 0 4 If,, c e non-copln unit vectos such tht c (, nd c e non-pllel, then pove tht the ngle etween nd is 4 FINAL STEP EXERCISE (SUBJECTIVE),, c e thee non-zeo vectos such tht ny two of them e non-colline If is colline with c nd c is colline with then pove tht 6c is null vecto In tingle A, B, C, D nd E e points on BC nd AC espectively such tht BD = DC nd AE = EC Let P e the point of intesection of AD nd BE Find BP/PE using vecto methods P, Q, R divide the sides BC, CA nd AB of ABC espectively in the tio : The thee lines AP, BQ nd CR enclose tingle XYZ Pove tht the tingles ABC nd XYZ hve the sme centoid 4 The medin AD of tingle ABC is isected t E nd BE is poduced to meet AC in F Pove y vecto method tht EF = 4 BF 5 In qudiltel PQRS, PQ,QR nd SP, M is the mid-point of QR nd X is 4 point of SM such tht SX SM Exmine 5 whethe P, X nd R e colline 6 If I e the cente of cicle inscied in tingle ABC, then the vecto sum BC IA CA IB ABIC 0 7 The position vecto of the points P nd Q e 5î 7 ĵ kˆ nd î ĵ 6kˆ espectively The vecto A î ĵ kˆ psses though the point P nd the vecto B î ĵ 4kˆ psses though the point Q A thid vecto î 7 ĵ 5kˆ intesect vectos A nd B Find the position vectos of the points of intesection 8 If O is the cicumcente nd O the othocente of tingle ABC, then pove tht (i) (ii) (iii) OA OB OC OO OA OB OC OO AO OB OC AO AP whee AP is the dimete though A of cicumcicle 9 If O e the cicumcente; G the centoid nd H the othocente of tingle ABC, pove tht O, G, H e colline nd tht G divides OH in the tio : 0 Find the eqution of the plne pssing though the point A(,, ) nd pependicul to the vecto 4î 7ĵ 4kˆ If PM e the pependicul fom the point P(,, ) to this plne, find its length Find the distnce of P(î ĵ kˆ ) fom the plne L which psses though the thee points Einstein Clsses, Unit No 0, 0, Vdhmn Ring Rod Plz, Viks Pui Extn, Oute Ring Rod New Delhi 0 08, Ph : 96905, 857
MV A(î ĵ kˆ ),B(î ĵ kˆ),c(î ĵ kˆ) Also find the position vecto of the foot of pependicul fom P on the plne L nd e unit vectos pependicul to ech othe c is nothe unit vecto inclined t n ngle to oth nd If c p( ) q( ) whee p, q R, then show tht 4 4 Let e unit vecto nd e non-zeo vecto not pllel to Find the ngles of the tingle, two sides of which e epesented y the vectos (â ) nd (â)â 4 Pove tht the lines t( nd s(c ) will intesect if c c 5 If c e given non-zeo scl nd A nd B e given non-zeo pependicul vecto, find the vecto X which stisfies the eqution A X B AX c nd 6 Given tht the vectos nd e pependicul to ech othe, find vecto v in tems of nd stisfying the eqution v 0, v nd v 7 Given the vecto AB nd AC c coincident with the sides of tingle ABC Find esolution (wt the sis, c ) of the vecto dwn fom the vetex B of the ABC nd coinciding with the ltitude BD 8 A tethedon hs vetices t O(0, 0, 0), A(,, ), B(,, ) nd C(,, ) Then the ngle etween 9 the fces OAB nd ABC will e cos 5 9 If,, c, d e the position vectos of the vetices of cyclic qudiltel ABCD, then show tht c d ( )(d ) c cd d ( (d 0 0 Find the distnce of the point B(î ĵ kˆ ) fom the line which is pssing though A(4î ĵ kˆ ) nd which is pllel to the vecto c (î ĵ 6kˆ ) Let 4î ĵ nd e two vectos pependicul to ech othe in the x-y plne Find ll the vectos in the sme plne hving pojections nd long nd? Pove tht In tingle ABC, pove using vecto methods cosa + cosb + cosc 4 A line mkes ngles,,, with the digonls of cue pove tht cos + cos + cos + cos = 4 Also pove tht the ngle etween two digonls of cue is cos 5 Let î ĵnd î kˆ Show tht the points of intesection of the lines nd isî ĵ kˆ 6,, c e thee non-copln unit vectos such tht ngle etween ny two is If c l m nc, then detemine l, m, n in tems of 7 Pove tht if l,m, n e thee non-copln vectos then [l l m n] ( ) m n l m n l m n 8 If,, c e thee non-copln vectos, then pove tht d d cd d ( (c d) ( ) [ c] [c] [c] Einstein Clsses, Unit No 0, 0, Vdhmn Ring Rod Plz, Viks Pui Extn, Oute Ring Rod New Delhi 0 08, Ph : 96905, 857
9 If,, c e non-copln vectos nd d is unit vecto, then find the vlue of (d)( (d)(c ) (cd)( ) independent of d MV 0 If p( q(c ) ( ) nd [ c], then find the vlue of p q [ c] ANSWERS SUBJECTIVE (INITIAL STEP EXERCISE) 7 î ĵ kˆ 4 m n 4 5 c î ĵ kˆ 6 6 7 ANSWERS SUBJECTIVE (FINAL STEP EXERCISE) 5 P, X, R e colline 7 ( +, 8, ) 8 0 units 9 ( î ĵ) 5 î ĵ 7kˆ + = 7 6 A, A, A ANSWERS (SINGLE CORRECT CHOICE TYPE) c 4 c 5 6 c 7 c 8 9 0 d c c 4 5 6 7 8 c 9 0 Einstein Clsses, Unit No 0, 0, Vdhmn Ring Rod Plz, Viks Pui Extn, Oute Ring Rod New Delhi 0 08, Ph : 96905, 857