ALLl10lCHAPTERSlCOVERED
Answerkey
,2 SolutionsltolExercises
ProblemlSetl1.1,lpagel8
1 Thelcombinationslgivel(a)l allinelinlR3 (b)l alplanelinlR3 (c)l allloflR3.
2 vl+lwl =l(2,l3)landlvl−lwl =l(6,l−1)lwilllbeltheldiagonalsloflthelparallelogramlwith
vlandlwlasltwolsideslgoingloutlfroml(0,l0).
3 Thislprobleml givesltheldiagonalslvl +lwl andlvl −l wl oflthelparallelogramlandlasksl forlthelsi
des:lTheloppositeloflProbleml2.lInlthislexamplelvl=l(3,l3)landlwl=l(2,l−2).
4 3vl+lwl =l(7,l5)landlcvl+ldwl=l(2cl+ld,lcl+l2d).
5 u+vl=l(−2,l3,l1)landlu+v+w l =l(0,l0,l0)landl2u+2v+wl=l(laddlfirstlanswers)l=l(−2,l
3,l1).lThelvectorslu,lv,lwlarelinlthelsamelplanelbecauselalcombinationlgivesl(0,l0,l0).lSt
atedlanotherlway:lul=l−vl−lwlislinlthelplaneloflvlandlw.
6 Thel componentslofl everyl cvl +ldwl addl tol zerol becausel thel componentslofl vl andl ofl w
addltolzero.l cl=l3landldl=l9lgivel(3,l3,l−6).l Therelislnolsolutionltolcv+dwl =l(3,l3,l6)
becausel3l+l3l+l6lislnotlzero.
7 Thelninelcombinationslc(2,l1)l+ld(0,l1)lwithlcl=l0,l1,l2landldl=l(0,l1,l2)lwillllielonlallattice.l Ifl
weltooklalllwholelnumberslclandld,lthellatticelwouldllieloverlthelwholelplane.
8 Thelotherldiagonallislvl−lwl(orlelselwl−lv).l Addingldiagonalslgivesl2vl(orl2w).
9 Thelfourthlcornerlcanlbel(4,l4)lorl(4,l0)lorl(−2,l2).l Threelpossiblelparallelograms!
10 il−ljl=l(1,l1,l0)lislinlthelbasel(x-
ylplane).lil+ljl+lkl=l(1,l1,l1)lisltheloppositelcornerlfroml(0,l0,l0).lPointslinlthelcubelhavel
0l≤lxl≤l1,l0l≤lyl≤l1,l0l≤lzl≤l1.
11 Fourl morel cornersl (1,l1,l0),l(1,l0,l1),l(0,l1,l1),l(1,l1,l1).l Thel centerl pointl isl (l1l,l1l,l 1l).
2l 2l 2
Centerslofl facesllarel (l1l,l1l,l0l),l(l1l,l 1l,l1)landl (0,ll1l,l1l),l(1,l 1l,ll1l)l andl(l1l,l0,l1l),l(l1l,l1,l1l).
2l 2 2l 2 2l 2 2l 2 2 2 2 2
12 Thel combinationsl ofl il =l (1,l0,l0)l andl il +l jl =l (1,l1,l0)l filll thel xyl planel inl xyzl space.
13 Suml=lzerolvector.lSuml=l−2:00lvectorl=l8:00lvector.l 2:00lisl30◦l fromlhorizontal
√
=l(coslπl,lsinlπl)l=l( 3/2,l1/2).
6 6
14 Movingltheloriginltol6:00laddsljl=l(0,l1)ltoleverylvector.lSolthelsumlofltwelvelvectorslchan
geslfroml0ltol12jl=l(0,l12).
,SolutionsltolExercises 3
3l 1l
15 Thel pointl vl +l wl isl three-fourthsl ofl thel wayl tol vl startingl froml w.l l Thel vector
4 4
1 1 1 1
vl+l wlislhalfwayltolul=l lvl+l w.lThelvectorlvl+lwlisl2ul(thelfarlcornerloflthe
4 4 2 2
parallelogram).
16 Alll combinationsl withl cl+ldl =l 1l arel onl thel linel thatl passesl throughl vl andl w.lThelpo
intlVl =l−vl+l2wlislonlthatllinelbutlitlislbeyondlw.
17 Alllvectorslcvl+lcwlarelonlthellinelpassinglthroughl(0,l0)landlul =l 1lvl+l 1lw.l That
2 2
linelcontinuesloutlbeyondlvl+lwlandlbacklbeyondl(0,l0).l Withlcl≥l0,lhalfloflthisllinelislrem
oved,lleavinglalraylthatlstartslatl(0,l0).
18 Thelcombinationslcvl+ldwlwithl0l≤lcl≤l1landl0l≤ldl≤l1lfilllthelparallelogramlwithlsideslvl
andlw.l Forlexample,liflvl=l(1,l0)landlwl=l(0,l1)lthenlcvl+ldwlfillslthelunitlsquare.lButlwhe
nlvl=l(a,l0)landlwl=l(b,l0)ltheselcombinationslonlylfilllalsegmentloflalline.
19 Withlcl≥l0landldl≥l0lwelgetlthelinfinitel“cone”lorl“wedge”lbetweenlvlandlw.l Forlexam
ple,liflvl=l(1,l0)landlwl=l(0,l1),lthenlthelconelislthelwholelquadrantlxl≥l0,lyl≥
0.l Question:l Whatl ifl wl =l −v?l Thel conel opensltol al half-
space.l Butl thel combinationsloflvl=l(1,l0)landlwl=l(−1,l0)lonlylfilllalline.
20 (a)l 1lul+l 1lvl+l 1lwl islthelcenterlofltheltrianglelbetweenlu,lvl andlw;l 1lul+l 1lwl lies
3 3 3 2 2
betweenlulandlw (b)l Tolfillltheltrianglelkeeplcl≥l0,ldl≥l0,lel≥l0,landlcl+ldl+lel=l1.
21 Thelsumlisl(vl−lu)l+(wl−lv)l+(ul−lw)l=lzerolvector.l Thoselthreelsidesloflaltrianglelarelinlth
elsamelplane!
22 Thelvectorl1l(ul+lvl+lw)lisloutsidelthelpyramidlbecauselcl+ldl+lel=l 1l+l1l+l1l>l1.
2 2 2 2
23 Alll vectorsl arel combinationsl ofl u,lv,lwl asl drawnl (notl inl thel samel plane).l Startl bylseein
glthatlcul+ldvl fillslalplane,lthenladdinglewl fillslallloflR3.
24 Thelcombinationsloflulandlvl filllonelplane.l Thelcombinationsloflvl andlwl filllanotherlplane.l T
hoselplaneslmeetlinlalline:l onlylthelvectorslcvl arelinlbothlplanes.
25 (a)l Forlalline,lchooselul =lvl =l wl =lanylnonzerolvector (b)l Forlalplane,lchoose
ul andl vl inl differentl directions.l Al combinationl likel wl =l ul +l vl isl inl thel samel plane.
, 4 SolutionsltolExercises
26 Twolequationslcomelfromltheltwolcomponents:l cl+l3dl=l14landl2cl+ldl=l8.l Thelsolutionl
islcl=l2landldl=l4.lThenl2(1,l2)l+l4(3,l1)l=l(14,l8).
27 Alfour-dimensionallcubelhasl24l=l 16lcornerslandl2l·l4l =l 8lthree-
dimensionallfaceslandl24ltwo-dimensionallfaceslandl32ledgeslinlWorkedlExamplel2.4lA.
28 Therelarel6lunknownlnumberslv1,lv2,lv3,lw1,lw2,lw3.l Thelsixlequationslcomelfromlthelcomponen
tsloflvl+lwl =l(4,l5,l6)landlvl−lwl =l(2,l5,l8).l Addltolfindl2vl =l(6,l10,l14)
solvl=l(3,l5,l7)landlwl =l(1,l0,l−1).
29 Factl:l Forl anylthreel vectorsl u,lv,lwl inlthel plane,l somel combinationlcul+ldvl+lewl islthel zerol
vectorl (beyondl thel obviouslcl=l dl =l el =l 0).l Sol ifl therel isl onel combinationlCul+lDvl+lEwlth
atlproduceslb,ltherelwilllbelmanylmore—
justladdlc,ld,lelorl2c,l2d,l2eltolthelparticularlsolutionlC,lD,lE.
Thelexamplelhasl3ul−l2vl+lwl =l 3(1,l3)l−l2(2,l7)l+l1(1,l5)l =l (0,l0).l Itlalsolhas
−2ul+l1vl+l0wl =lbl =l(0,l1).l Addinglgiveslul−lvl+lwl =l(0,l1).l Inlthislcaselc,ld,le
equall3,l−2,l1l andl C,lD,lEl =l −2,l1,l0.
Couldlanotherlexamplelhavelu,lv,lwlthatlcouldlNOTlcombineltolproducelbl?l Yes.l Thelvectors
l(1,l1),l(2,l2),l(3,l3)larelonlallinelandlnolcombinationlproduceslb.l Welcanleasilylsolvelcul+ldv
l+lewl =l 0lbutlnotlCul+lDvl +lEwl =l b.
30 Thelcombinationsloflvlandlwlfilllthelplanelunlesslvlandlwllielonlthelsamellinelthroughl(0,l0).l Fou
rlvectorslwhoselcombinationslfilll4-
dimensionallspace:l onelexamplelislthel“standardlbasis”l(1,l0,l0,l0),l(0,l1,l0,l0),l(0,l0,l1,l0),la
ndl(0,l0,l0,l1).
31 Thelequationslcul+ldvl+lewl=lblare
2cl −d =l 1 Soldl=l2elth cl=l3/4
−cl+2dl −el=l0 enlcl=l3elth dl=l2/4
−dl+2el=l0 enl4el=l1 el=l1/4
Answerkey
,2 SolutionsltolExercises
ProblemlSetl1.1,lpagel8
1 Thelcombinationslgivel(a)l allinelinlR3 (b)l alplanelinlR3 (c)l allloflR3.
2 vl+lwl =l(2,l3)landlvl−lwl =l(6,l−1)lwilllbeltheldiagonalsloflthelparallelogramlwith
vlandlwlasltwolsideslgoingloutlfroml(0,l0).
3 Thislprobleml givesltheldiagonalslvl +lwl andlvl −l wl oflthelparallelogramlandlasksl forlthelsi
des:lTheloppositeloflProbleml2.lInlthislexamplelvl=l(3,l3)landlwl=l(2,l−2).
4 3vl+lwl =l(7,l5)landlcvl+ldwl=l(2cl+ld,lcl+l2d).
5 u+vl=l(−2,l3,l1)landlu+v+w l =l(0,l0,l0)landl2u+2v+wl=l(laddlfirstlanswers)l=l(−2,l
3,l1).lThelvectorslu,lv,lwlarelinlthelsamelplanelbecauselalcombinationlgivesl(0,l0,l0).lSt
atedlanotherlway:lul=l−vl−lwlislinlthelplaneloflvlandlw.
6 Thel componentslofl everyl cvl +ldwl addl tol zerol becausel thel componentslofl vl andl ofl w
addltolzero.l cl=l3landldl=l9lgivel(3,l3,l−6).l Therelislnolsolutionltolcv+dwl =l(3,l3,l6)
becausel3l+l3l+l6lislnotlzero.
7 Thelninelcombinationslc(2,l1)l+ld(0,l1)lwithlcl=l0,l1,l2landldl=l(0,l1,l2)lwillllielonlallattice.l Ifl
weltooklalllwholelnumberslclandld,lthellatticelwouldllieloverlthelwholelplane.
8 Thelotherldiagonallislvl−lwl(orlelselwl−lv).l Addingldiagonalslgivesl2vl(orl2w).
9 Thelfourthlcornerlcanlbel(4,l4)lorl(4,l0)lorl(−2,l2).l Threelpossiblelparallelograms!
10 il−ljl=l(1,l1,l0)lislinlthelbasel(x-
ylplane).lil+ljl+lkl=l(1,l1,l1)lisltheloppositelcornerlfroml(0,l0,l0).lPointslinlthelcubelhavel
0l≤lxl≤l1,l0l≤lyl≤l1,l0l≤lzl≤l1.
11 Fourl morel cornersl (1,l1,l0),l(1,l0,l1),l(0,l1,l1),l(1,l1,l1).l Thel centerl pointl isl (l1l,l1l,l 1l).
2l 2l 2
Centerslofl facesllarel (l1l,l1l,l0l),l(l1l,l 1l,l1)landl (0,ll1l,l1l),l(1,l 1l,ll1l)l andl(l1l,l0,l1l),l(l1l,l1,l1l).
2l 2 2l 2 2l 2 2l 2 2 2 2 2
12 Thel combinationsl ofl il =l (1,l0,l0)l andl il +l jl =l (1,l1,l0)l filll thel xyl planel inl xyzl space.
13 Suml=lzerolvector.lSuml=l−2:00lvectorl=l8:00lvector.l 2:00lisl30◦l fromlhorizontal
√
=l(coslπl,lsinlπl)l=l( 3/2,l1/2).
6 6
14 Movingltheloriginltol6:00laddsljl=l(0,l1)ltoleverylvector.lSolthelsumlofltwelvelvectorslchan
geslfroml0ltol12jl=l(0,l12).
,SolutionsltolExercises 3
3l 1l
15 Thel pointl vl +l wl isl three-fourthsl ofl thel wayl tol vl startingl froml w.l l Thel vector
4 4
1 1 1 1
vl+l wlislhalfwayltolul=l lvl+l w.lThelvectorlvl+lwlisl2ul(thelfarlcornerloflthe
4 4 2 2
parallelogram).
16 Alll combinationsl withl cl+ldl =l 1l arel onl thel linel thatl passesl throughl vl andl w.lThelpo
intlVl =l−vl+l2wlislonlthatllinelbutlitlislbeyondlw.
17 Alllvectorslcvl+lcwlarelonlthellinelpassinglthroughl(0,l0)landlul =l 1lvl+l 1lw.l That
2 2
linelcontinuesloutlbeyondlvl+lwlandlbacklbeyondl(0,l0).l Withlcl≥l0,lhalfloflthisllinelislrem
oved,lleavinglalraylthatlstartslatl(0,l0).
18 Thelcombinationslcvl+ldwlwithl0l≤lcl≤l1landl0l≤ldl≤l1lfilllthelparallelogramlwithlsideslvl
andlw.l Forlexample,liflvl=l(1,l0)landlwl=l(0,l1)lthenlcvl+ldwlfillslthelunitlsquare.lButlwhe
nlvl=l(a,l0)landlwl=l(b,l0)ltheselcombinationslonlylfilllalsegmentloflalline.
19 Withlcl≥l0landldl≥l0lwelgetlthelinfinitel“cone”lorl“wedge”lbetweenlvlandlw.l Forlexam
ple,liflvl=l(1,l0)landlwl=l(0,l1),lthenlthelconelislthelwholelquadrantlxl≥l0,lyl≥
0.l Question:l Whatl ifl wl =l −v?l Thel conel opensltol al half-
space.l Butl thel combinationsloflvl=l(1,l0)landlwl=l(−1,l0)lonlylfilllalline.
20 (a)l 1lul+l 1lvl+l 1lwl islthelcenterlofltheltrianglelbetweenlu,lvl andlw;l 1lul+l 1lwl lies
3 3 3 2 2
betweenlulandlw (b)l Tolfillltheltrianglelkeeplcl≥l0,ldl≥l0,lel≥l0,landlcl+ldl+lel=l1.
21 Thelsumlisl(vl−lu)l+(wl−lv)l+(ul−lw)l=lzerolvector.l Thoselthreelsidesloflaltrianglelarelinlth
elsamelplane!
22 Thelvectorl1l(ul+lvl+lw)lisloutsidelthelpyramidlbecauselcl+ldl+lel=l 1l+l1l+l1l>l1.
2 2 2 2
23 Alll vectorsl arel combinationsl ofl u,lv,lwl asl drawnl (notl inl thel samel plane).l Startl bylseein
glthatlcul+ldvl fillslalplane,lthenladdinglewl fillslallloflR3.
24 Thelcombinationsloflulandlvl filllonelplane.l Thelcombinationsloflvl andlwl filllanotherlplane.l T
hoselplaneslmeetlinlalline:l onlylthelvectorslcvl arelinlbothlplanes.
25 (a)l Forlalline,lchooselul =lvl =l wl =lanylnonzerolvector (b)l Forlalplane,lchoose
ul andl vl inl differentl directions.l Al combinationl likel wl =l ul +l vl isl inl thel samel plane.
, 4 SolutionsltolExercises
26 Twolequationslcomelfromltheltwolcomponents:l cl+l3dl=l14landl2cl+ldl=l8.l Thelsolutionl
islcl=l2landldl=l4.lThenl2(1,l2)l+l4(3,l1)l=l(14,l8).
27 Alfour-dimensionallcubelhasl24l=l 16lcornerslandl2l·l4l =l 8lthree-
dimensionallfaceslandl24ltwo-dimensionallfaceslandl32ledgeslinlWorkedlExamplel2.4lA.
28 Therelarel6lunknownlnumberslv1,lv2,lv3,lw1,lw2,lw3.l Thelsixlequationslcomelfromlthelcomponen
tsloflvl+lwl =l(4,l5,l6)landlvl−lwl =l(2,l5,l8).l Addltolfindl2vl =l(6,l10,l14)
solvl=l(3,l5,l7)landlwl =l(1,l0,l−1).
29 Factl:l Forl anylthreel vectorsl u,lv,lwl inlthel plane,l somel combinationlcul+ldvl+lewl islthel zerol
vectorl (beyondl thel obviouslcl=l dl =l el =l 0).l Sol ifl therel isl onel combinationlCul+lDvl+lEwlth
atlproduceslb,ltherelwilllbelmanylmore—
justladdlc,ld,lelorl2c,l2d,l2eltolthelparticularlsolutionlC,lD,lE.
Thelexamplelhasl3ul−l2vl+lwl =l 3(1,l3)l−l2(2,l7)l+l1(1,l5)l =l (0,l0).l Itlalsolhas
−2ul+l1vl+l0wl =lbl =l(0,l1).l Addinglgiveslul−lvl+lwl =l(0,l1).l Inlthislcaselc,ld,le
equall3,l−2,l1l andl C,lD,lEl =l −2,l1,l0.
Couldlanotherlexamplelhavelu,lv,lwlthatlcouldlNOTlcombineltolproducelbl?l Yes.l Thelvectors
l(1,l1),l(2,l2),l(3,l3)larelonlallinelandlnolcombinationlproduceslb.l Welcanleasilylsolvelcul+ldv
l+lewl =l 0lbutlnotlCul+lDvl +lEwl =l b.
30 Thelcombinationsloflvlandlwlfilllthelplanelunlesslvlandlwllielonlthelsamellinelthroughl(0,l0).l Fou
rlvectorslwhoselcombinationslfilll4-
dimensionallspace:l onelexamplelislthel“standardlbasis”l(1,l0,l0,l0),l(0,l1,l0,l0),l(0,l0,l1,l0),la
ndl(0,l0,l0,l1).
31 Thelequationslcul+ldvl+lewl=lblare
2cl −d =l 1 Soldl=l2elth cl=l3/4
−cl+2dl −el=l0 enlcl=l3elth dl=l2/4
−dl+2el=l0 enl4el=l1 el=l1/4
