Inspiration and formalism

Inspirtion n formlism Answers Skills hek P(, ) Q(, ) PQ + ( ) PQ A(, ) (, ) grient ( ) + Eerise A opposite sies of regulr hegon re equl n prllel A ED i FC n ED ii AD, DA, E, E n FC No, sies of pentgon re prllel the vetors in the igrm re ll istrit. No, sine no vetors in the igrm re prllel n equl in length Eerise ( + ) + + ( + ) (+) + + C + (+) AF + C AF + FE AE e f AD + ED FE + ED FD + FE AF FE FE + FA FE + EO FO A (AD + E) AD + E AO + OE AE FC + C OF + FE OE CD ED AF + A CF + FA + A C (other nswers re possile in this question) i AC u v ii H uw u u w iii CE vu wu vwu v iv AFwv C i A D C + os AC ˆ 7 ii re ACD sin 7.7 sq. units u v + u v + u v + u ( u) + (u v) u + u v vu u v u + v u + v Eerise C A(, ) C(, ) I(7, 8) i A AC. ii AE A CI 7. 8. iii CD CI i F A + AF + i +. j. ii CH C + H. + i +. j iii DG DF + FG + i + j O OA + A +.. OD OC + CD + OE OA + AE +.. OF OA + AF + OG OA + AG + 7 Ofor Universit Press : this m e reproue for lss use solel for the purhser s institute Worke solutions: Chpter

P(,, ) Q(,, ) OP OQ PQ i jk A AD AE AG A + C + CG e D A + AD + f H D + DH P(, ) Q(, 7) R(, ) OP OQ 7 OR M(, ) N(, ) QR 7 MN QR MN (QED) Eerise D u u u u u + u u u u u (QED) u v w u v w u vw u v w u v w u v w u vw u v w u v w u vw u v w uv w QED Ofor Universit Press : this m e reproue for lss use solel for the purhser s institute u u u u u u u u u u u u u u u (QED) u v u v u v uv u v u v u v u v u v uv (QED) u v u v e u u u u u u u u u u u u u uu (QED) u u u u f u u u u u (QED) g (QED),, 7 u + (v + u) u + v (u v) + (v u) u v + v u u + v uv v u uvvu u v i j i j + β i j (i j) + β(i j) i j + β i j (i j) + β(i j) i j β, β β i j Eerise E v v v v v v v v Worke solutions: Chpter u

v v ± or v or v v or v or v u i j u v u u v u or u v u or u u v u or u i j u w u i j u i j u t u j or t i t i j v 7 u v u kv k v v v v v 8 u is in the sme iretion s v (QED) v v v v u u v v v v v v u hs mgnitue (QED) u u mv m v v v v mv v mv m v v mv m v v v v u m (QED) Eerise F A(, ) (, ) C(, ) D(, ) A AC AD D A + AD + A(, ) (, ) A + ( ) A A M(, ) M is the mipoint of A Let P(, ) AP P AP P ( ), ( ) P,, 8 Let Q (, ), AQ AQ Q Q ( ), ( ) + 8 + Q(, ) A < PQ A P Q PQ hs greter mgnitue thn A P(, ) Q(, ) R(, ) PQ PR PR PQ PQ n PR re olliner P, Q, R re olliner (QED) A(, ) (, ) C(, ) A AC + + k( ) + k( + ) + ± + 8, ± + + AC k A Ofor Universit Press : this m e reproue for lss use solel for the purhser s institute Worke solutions: Chpter

Worke solutions: Chpter Ofor Universit Press : this m e reproue for lss use solel for the purhser s institute S(, ) U(, ) N(, ) SU SN SN ksu S, U, N re not olliner the form tringle (QED) P(, ) Q(, ) R(e, f ) PQ PR e f f e f e PR f f f PR f PQ PR k PQ P, Q, R re olliner points 7 A sin sin os os sin os sin os os sin os os os sin os os os sin os os Therefore for n vlue of, A is olliner with sin os A os sin os os os Eerise G u i + j + k v i + j k u + v i + j k u i j k u v 8i + j + k (i + j k) i + 8j + k (u v) (i + j + k) i j 8k + + + ( ) 7 e f g + + h A(,, ) (,, ) C(,, ) A AC A AC C AC A + i j k v u A(,, ) C(,, ) D(,, ) G(,, ) AC AD CG A AC AD O OA + A AE CG OE OA + AE F CG OF O + F AH AD CG OH OA + AH 7

Eerise H PQ + r + λ, + λ + r + + λ, λ, + λ + + + (,, ) or eg use λ,, (,, ) (,, ) (,, ) (other solutions re possile) λ + λ At P, P oes not lie on L (QED) r + eg use k, (,, ) (,, ) iretion of line i j u i j i j Eerise I u v. os u (v) u v os (π θ) u v os (π θ) u v os θ (u v) (u) v u v os (π θ) u v os (π θ) u v os θ (u v) u (v) (u v) (u) v (QED) C Let the length of the sies e A C C AC os os A C + C AC + A AC os CA C os sin sin AC C os sin sin A i A AC os θ ± Angle etween u n u 7 u u u u os u v u u v u v u v os C Are sin θ sin θ os θ ± (QED) u + v if u + v u v the prllelogrm is retngle n the ngle etween u n v is (QED) 8 igonl AG 7 + + O OC os ÔC 7 O OC OA OE OA OE Eerise J os AÔE u v + u v + 7 A(,, ) (,, ) C(,, ) A C A C + AC C + 8 AC 7 A Ofor Universit Press : this m e reproue for lss use solel for the purhser s institute Worke solutions: Chpter

(,, ) A(,, ) (,, ) C(,, ) D(,, ) E(,, ) F(,, ) G(,, ) OF OG AF G AF G D O A C AC OA O OC OD Volume OE 8 OE A(,, ) (,, ) C(,, ) D(,, ) E(,, ) EA E os AÊ + AÊ 8.º Eerise K EA EA E 8 u i j v i + j u v u v os θ θ u i j + k v i j + k u v + + u v os θ θ. A(,, ) (,, ) C(,, ) A AC A AC os θ θ C C AC + 8 + 7 C AC 7 os θ 7 7 θ. u v > ( ) + ( ) > + > When ( ) ( + ) > or < or > u sin α i os α j + k v os α i sin α j k u v sin α os + os α sin α sin α u sin α + os α + u v os α + sin α + u os θ sin sin sin os sin α. α.78,.7,.7,.,., 8.,.8, 7.7 α.87,.,.,.,.,.,.8,. sin α < sin α < os θ < θ is otuse (QED) Let + + + + + + + + ( + ) ( + ) ( + ) ( + ) + ( + ) ( + ) + ( + ) ( + ) + + + + + + + + + + + ( + + ) + ( + + ) + ( + + ) + ( + + ) + + + (QED) Eerise L u i j k v i j + k w i k u v i j k v w i + j + k u (v w) j + k Ofor Universit Press : this m e reproue for lss use solel for the purhser s institute Worke solutions: Chpter

(u v) w i j + k e u w i + k f (u v) (u w) i + j + k g (u + v) (u w) (i j) (j) k A(,, ) (,, ) C(,, ) E(,, ) AD C OD OA + AD D (,, ) Volume (A AD) AE A AD Volume u. units u v (u v u v ) i + (u v u v ) j + (u v u v ) k u i j k v i j + k w i k From qn., u (v w) j + k (u v) w i j + k (v u (v w) (u v) w u v u ) i (v u v u ) j (v not ssoitive (QED) u v u ) k v u (v w u n w v u v u )i + (v u v u ) j + (v w u + w u + w u n u v u ) k w v + w v + w v u v v u (QED) u v (u v u v ) i + (u v u v ) j u v + w + (u v u v ) k u (v + w) u v + w w (u v) [w (u v u v ) w (u v u v )] i u v + w + [w (u v u v ) w (u v u v )] j + [w (u v u v ) w (u v u v )] k [u (w v + w v ) v (w u + w u )] i + [u (w v + w v ) v (w u + w u )] j + [u (w v + w v ) v (w u + w u )] k [u (v + w ) u (v + w )] i + [u (v + w ) u (v + w )] j + [u (v + w ) u (v + w )] k [(u v u v ) + (u w u w )] i + [(u v u v ) + (u w u w )] j + [(u v u v ) + (u w u w )] k [u (w v ) v (w u )] i (u v u v ) i + (u v u v ) j + [u (w v ) v (w u )] j + (u v u v ) k + [u (w v ) v (w u )] k w (u v) w n u v re olliner + (u w u w ) i + (u w u w ) j + (u w u w ) k A(,, ) (, 7, ) C(,, ) u (v + w) u v + u w (QED) uv uv u A AC OD OA + AD u v u uv uv u uv uv u D (,, ) u u v u u v + u u v u u v + u u v u u v A u v u (QED) C u v uv uv (λ u) v u v uv uv u v uv uv A C 8 uv uv re λ uv uv A C + + λ (u v) uv uv 8 7 (λu) v λ (u v) (QED) Ofor Universit Press : this m e reproue for lss use solel for the purhser s institute Worke solutions: Chpter 7

Worke solutions: Chpter Ofor Universit Press : this m e reproue for lss use solel for the purhser s institute 8 Eerise M A(,, ) (,, ) C(,, ) A AC r + + + α + β () + α () α β () From () α From () β + ( ) β + 7 su in (): + ( + 7) 8 + 8 + (other forms re possile) P(,, ) n i + j k r n n + + eg (,, ) (,, ) (,, ) (,, ) n P(,, ) line psses through A(,, ) n hs iretion AP, n + + n t + t t (,, ) lies in the plne n re vetors in the plne n h h (,, ) A(,, ) (,, ) C(,, ) V(,, ) A AV r + α + β C CV n 8 8 8 + 8 V + r

e r + λ,, λ λ λ, Eerise N r + λ Metho u v os θ θ 7 Metho m m tn μ tn β θ β θ β θ ute ngle 7 u v u v + 8 + u v 7 os θ 7 θ. r ( θ ) i + ( + θ ) j + ( θ ) k r ( + β ) j + ( + β ) k u v u v os θ u m u θ 7. os θ u v v m u v + m m m v + m u os θ v + mm + m + m tn (α β ) tn tn m m + tn tn + mm se (α β ) + tn (α β ) + m m ( ) ( + mm ) + ( ) + mm m m + mm + mm + mm + m mm + m + mm + m + m + m m + mm + ( ) + m m + mm os(α β ) + mm + m + m os θ os(α β ) sine θ α β (> sine α > β) Eerise O r ( λ) i + ( λ) j + ( + λ) k u + n sin θ u n u n θ u r ( θ β ) i + ( θ + β ) j + (θ + β ) k n 8 u n + 8 + sin θ 8 θ. + m r ( θ + β ) i + ( β ) j + ( θ β ) k n os θ θ.8 m n + + 8 Ofor Universit Press : this m e reproue for lss use solel for the purhser s institute Worke solutions: Chpter

A(,, ) (,, ) C(,, ) D(,, ) A AC n 7 7 7 7 AD r + λ + λ λ + λ u n u n 7 7 sin θ 7 7 θ. k k (k ) k + + k k k k u k k n k k If the line n plne re prllel, u n so k (k ) + k + k k + k k(k ) k or Eerise P r i + k + λ(i j + k) P( + λ, λ, + λ) + + + λ λ + ( + λ) λ P(,, ) λ + λ, λ, λ + λ λ + λ λ λ (,, ) k, k, k r ( θ + β ) i + ( β ) j + ( θ β ) k θ + β β θ β k θ + β k + θ β k β k + β k θ β k + θ + β k., θ., β. (.8,.8,.) r ( + λ) i + ( + λ) j + ( + λ) k, u n u n u n n re perpeniulr the line prllel to the plne (QED) (,, ) lies in the plne r + λ lies in the plne r λi + λ j + ( + λ) k (other nswers re possile) P(,, ) + n line through p perpeniulr to the plne: r + λ point of intersetion, I( + λ, + λ, λ) ( + λ) + + λ ( λ) + λ + + λ + λ λ λ. I(.8,., ) PI ( 8. ) + (. ) + 8. 8 istne or. Eerise Q + () () + +, su in () su in () + interset t (,, ) r ( + λ) i + ( + λ) j + ( λ) k, + P( + λ, + λ, λ) + λ + λ n + λ λ + λ λ λ Equtions re inonsistent no point of intersetion Ofor Universit Press : this m e reproue for lss use solel for the purhser s institute Worke solutions: Chpter

+ 8 () + () () + () () + () + +. + +. + n ( k) λ (k ) + λ k + kλ u k n n u re olliner k + λ, + λ, λ ( λ) ( + λ) + ( λ) λ + λ λ λ 8 λ 8,, L : +, iretion, u L : iretion, v v u L n L re prllel (QED) A(,, ) n (,, ) re points in the plne A n + 7 + + n n u u n + u n n re perpeniulr the line is prllel to the plne. (,, ) lies on the line + + + + () (,, ) lso lies in the plne the plne ontins the line (QED) Eerise R + + () () () () + + () () () () + + () () () From (),, + + () () () + () + + () su in () n () + () + () () n () re inonsistent no ommon point (QED) + + () () () + + () ( ) + + () or if, + either + or If, either point of intersetion (,, ) or interset in stright line. if equtions () n () re the sme + + + + or ll plnes the sme or n not oth n interset in stright line. if, either point of intersetion (,, ) or interset in plne or interest in line. the plnes lws hve t lest one point in ommon. For stright line, ut,, not ll equl to or n n oth equl to + π + π + 7 π π n π re prllel ut not oinient, π intersets eh of the other plese in stright line ut the plnes hve no point in ommon. Ofor Universit Press : this m e reproue for lss use solel for the purhser s institute Worke solutions: Chpter

+ + () () + () + () + () () + () () () () + () 7 7,, (,, ) + + () () + () + + () () + () + (k + ) + k () For no ommon point, k + 8 k + k 7 Verif igrms Eerise S A t t t t A(, ) (, ) V A V os θ V V A V A V t A t t t t 7 t θ 7. A ( t) + ( 7t) This is minimum when t. hours OP ( + t) i + ( t) j + (t ) k, t > t, P(,, ) Crtesin equtions: + + ( ) i + + + t + t + t t t ii (,, ) iii P P istne + + t OQ t t > + t t t t t i PQ t + t 8t + t t + t t + PQ ( t t ) + ( 8t ) + ( t t + ) This is minimum when t.887. (sf) ii P(.,.,.88) Q(.,.8,.) 8 e i k( ) n re non olliner (QED) ii Q is not moving in stright line (,, ) C(,, ) OP C(,, ) G(,, ) OQ D(,, ) G(,, ) OR PQ PR n + + or + + Ofor Universit Press : this m e reproue for lss use solel for the purhser s institute Worke solutions: Chpter

Worke solutions: Chpter Ofor Universit Press : this m e reproue for lss use solel for the purhser s institute E(,, ) (,, ) OS A(,, ) E(,, ) OT A(,, ) (,, ) OU T U P Q R S V S, T, U lie in the plne PQR PQ QR RS ST TU UP ll sies re of length os P os Q os R os S os T os U ll ngles re PQRSTU is regulr hegon Are prllelogrm PQVU PQ PU re PQVU re hegon OF n OF is perpeniulr to the plne PQR (QED) e Generl point on OF is (λ, λ, λ) + + λ + λ + λ λ I,, f IF istne IF Review eerise u v u v u v u + v u + v 8 u i j + k i + j i + k i j k u α + β + γ α + β + γ α + β + r () () () β + γ () α γ () β γ () β γ () () () γ γ γ, β 7, α u + 7 + unit vetor ± ± u os α os β i + sin α os β j + sin β k u os α os β + sin α os β + sin β os β (os α + sin α) + sin β os β + sin β u u is unit vetor u i + tn α j v tn β i + j os r u v u v tn tn tn tn + + + tn tn se se + (tn β + tn α) os α os β sin β os α + sin α os β

Worke solutions: Chpter Ofor Universit Press : this m e reproue for lss use solel for the purhser s institute os γ sin (α + β) os + γ (α + β ) α + β + γ + i + () () os θ os θ + + () () os(π θ) θ os(π θ) + + os θ + + + + os θ ( os θ) + os θ 8 8 os θ os θ os θ ( θ ) sin θ os θ sin θ 7 r (i + j + k) + α (i j + k) + β ( j k) + α, α + β, + α β r (i k) + α (i + k) + β (j) α, β, + α 8 + λ + μ () λ + μ () From () μ λ Su in () + λ 8λ λ λ α P(, ) Review eerise u v os θ u v u v os θ + θ 8 (nerest egree) A AC r α β,, α β n or A AC r + α + β α + β, α, + α β n + or r (i + j k) r r (i j) + A AC n + + A n

Worke solutions: Chpter Ofor Universit Press : this m e reproue for lss use solel for the purhser s institute A n + e f n g + + () + + () () + Let λ, λ, λ λ λ Line of intersetion is r λ n + + or u v os θ u v u v θ 8. r i + j + k + λ(i + j + k) r i + j + k + α (i + j + k) + λ + α () + λ + α () + λ + α () From () λ α Su in () + + α α, λ Chek in () + () + () 7 7 L n L re onurrent point of intersetion is (,, 7) r i + j + 7k + α(i + j + k) + β(i + j + k) 7 + + () () () () + () () + () + 7 () + () () (),,, (,, ) is the point of intersetion of the plnes represente the equtions. 8 + + () () () + () + () () () + ( ) () + + () () () ( ) ( ) + + + 8 + + +, +, equtions () n () eome + + For non-unique solution,, the plnes represente the equtions interset in line.