HKCWCC dvned evel Pure Mhs. / -D Co-Geomer Three Dimensionl Coordine Geomer. Coordine of Poin in Spe Z XOX, YOY nd ZOZ re he oordine-es. P,, is poin on he oordine plne nd is lled ordered riple. P,, X Y O Y. Seion Formul X Z e P,, divides he poin in he rio P PB s r s r s r Therefore, P,,,,. r s r s r s Emples r s e P,, nd Q, -8, 8 e wo poins. Find he poin R whih divides he line segmen joining P nd Q in he rio.. Disne Formul e,, nd B,,, hen B.. Direion Cosines of Srigh ine Definiion. i Two srigh lines re // if nd onl if oplnr nd do no inerse. ii Two srigh lines re lled skew lines if nd onl if non-oplnr nd do no inerse. Z Definiion. Refer o he digrm P,, i Direion ngles α, β, γ. ii Direion osines of he direed lines OP [osα,osβ, osγ] [ l, m, n] α O γ β Y Theorem. e P,, nd P,, e wo poins. The direion osines of he direed line P P re osα osβ d d X osγ d Pge
HKCWCC dvned evel Pure Mhs. / -D Co-Geomer where d Proof e P P i j k P P i P P i osα P P i osα P P i d Similrl, for P P j osβ nd osγ P P j d P P k P P k d Noe Sudens should verif h os α os β os γ 5. Direion Rios of Srigh ine The direion of srigh line n e epressed s he rio of hree numers h re proporionl o he ul direion osines of he line. Suh numers re lled direion rios, nd used [λ µ ν ] o denoe he direion rios. Th is λ µ ν os α os β os γ. Noe Unlike he direion osines, he posiive direion of srigh line nno e deermined mens of direion rios. Theorem 5. The direion rios of he srigh. line pssing hrough he poins P,, nd P,, is defined s ] or ] [ [ The proof refers o eook p.5. Theorem 5. If he direion rios of srigh. line is [ ], hen he direion osines of he line is Clss Eerises osα, osβ, osγ ± ± ± Desrie he ehvior of he lines wih os α ; os α os β.. ngles eween Two Srigh ines. Theorem. If θ is he ngle eween wo direed lines l nd l, whose direion osines re l [osα, osβ, os ] nd l [osα,osβ, os ] γ γ hen osθ osα osα osβ osβ osγ osγ Proof We le OP nd OQ e wo uni veors in he sme direion s l nd l, respeivel, hen OP osα i osβ j osγ k nd OQ osα i osβ j osγ k Pge
HKCWCC dvned evel Pure Mhs. / -D Co-Geomer Hene, OP OQ os θ osα osα osβ osβ osγ osγ OP OQ Clss Eerises If θ is he ngle eween wo direed lines l nd l, whose direion rios re l [ ] nd l [ ], prove h osθ. Noe If he direion rios of wo srigh lines e ] nd ] respeivel. [ [ If wo direed lines re // if nd onl if. If he wo direed lines re perpendiulr if nd onl if 7. Equion of Srigh ines srigh line m e regrded s he inerseion of wo non-prllel plnes. Sine he equion of plne is liner equion B C D, hen he equion of srigh line n e wrien s B C D B C D whih is lled he generl form of srigh line. Smmeri Form When he direion rios of srigh line psses hrough he poin P,, re [ ], he direion rio of he srigh line m e lso e epressed s [ ], where,, is n poin on he srigh line. Then we hve, or This is lled he smmeri form of he srigh line. Clss Eerises Find he equion of he srigh line psses hrough he poin, -, nd prllel o he line segmen joining he poins, 5, nd B-,, 5. Find he equion of he line joining he poins,, - nd B,, -. Show h B mees he - plne in poin of inerseion of C, 7, - nd D-,,. B Prmeri form. The equion of srigh line psses hrough he poin,, nd wih he direion rios [ ] is. If we se s, hen we hve Pge
HKCWCC dvned evel Pure Mhs. / -D Co-Geomer Pge where is prmeer. This is lled he prmeri form of he srigh line. Clss Eerises Find he direion rios of he line. Trnsform he equion of he line 5 in smmeri form. Find he prmeri form of srigh line whih psses hrough he poin,, nd prllel o he srigh line. Show lso h his line is perpendiulr o he srigh line If he foo of he perpendiulr form he poin P, 7, -9 o he line is Q, find he oordine of Q. Hene, find he perpendiulr disne from P o. 5 Find he perpendiulr disne of he poin,, from he line 5. Consider he wo srigh lines nd. Find he poin of inerseions of nd. Find lso he ue ngle eween nd. 7 Consider he srigh line, where R. I is given h he line inerses B produed, where -,, - nd B,,. Find he lengh of he projeion of he line segmen B ono he line. 8 Consider he poins 7,, 5, B, -, nd C, -, 9 nd D-, -9, 7. Find he equions of he lines hrough B nd CD. Find he poin of inerseion of lines hrough B nd CD. Find lso he ngle eween hese wo lines. Find he lengh of he projeion of B on he line hrough CD. C Two-Poin Form The equion of he srigh line joining he poins P,, nd P,, is Noe Three poins P,,, P,, nd P,, re olliner if nd onl if
HKCWCC dvned evel Pure Mhs. / -D Co-Geomer 8. Equion of Plne The norml form or perpendiulr form The norml of plne is defined s he srigh line drwn from he origin O perpendiulr o he plne. nd he lengh of he norml is he disne eween he origin nd he foo of he perpendiulr of he norml p. Theorem 8. If he D.R. of he norml of he plne is [osα, osβ, osγ] nd p is he lengh of he norml, hen he equion of he plne is osα osβ osγ - p. Proof e P,, e n poin on he plne nd he oordine N of Q is osα, osβ, osγ Q OP OQ [ p osα i osβ j osγ k ] iposα jposβ kposγ pos α p osα p osβ osβ p osγ osγ os α osβ osγ p Q p P,, B The Generl Form The generl form of plne equion n e epressed s B C D nd ompred wih he equion os α osβ osγ p we hve he following relionship. osα osβ osγ p. I follows h [ B C] is he direion rios of he norml o he plne B C D B C D. ± ± B Hene osα, osβ nd osγ B C B C The hoie of ± sign is ording o he following rules If D, he plne does no pss hrough origin nd p >. Hene he sign of ± is ken s he opposie sign of D. When D nd C. The sign ± is ken he sme s C. When D, C nd B. The sign ± is ken he sme s B. d When D, C, B nd. The sign ± is ken he sme s. Theorem 8. The norml form of he equion of he plne B C D is B C D ± B C ± C B C where he hoie of ± sign is ording o he ove rule. Clss Eerises B reduing he equion o is norml form, find he direion osines of he norml Pge 5
HKCWCC dvned evel Pure Mhs. / -D Co-Geomer nd he perpendiulr disne from he origin o he plne. [D.C. /, -/, / nd p ] Find he signed disne from he poin,, o he plne. e π. Find he equion of wo plnes whih re prllel o π nd disne unis from he poin P,, [ nd 7 ] Theorem 8. e he equion of wo plnes e π B C D nd π B C D, hen If π // π, hen B C. B C If π π, hen B B C C C The perpendiulr disne eween poin nd plne Theorem 8. The perpendiulr disne eween poin P,, nd plne Clss Eerises π B C D is d B Find he perpendiulr disne eween wo prllel plnes B C D 7 π π 5 [ Find he lengh of projeion of he line segmen joining,, nd B,, ono he plne C ] π [ ] D ngles eween wo plnes The ngle eween wo plnes is defined s he ngle eween he non-direed norml lines of he wo plnes. Theorem 8.5 Clss Eerises If θ is he ngle eween he plnes π B C D nd π B C D, hen osθ ± B C B B C C Find he equion of he plne pssing hrough he poin P, -, nd Q,, nd mking n ngle wih he plne π. [ ± ] B C N.B. If one of he vriles,, in he equion B C D is missing, hen i B D is prllel o he -is or perpendiulr o he XY-plne. ii B C D is prllel o he -is or perpendiulr o he YZ-plne. iii C D is prllel o he -is or perpendiulr o he XZ-plne. If wo of he vriles,, in he equion B C D is missing, hen Pge
HKCWCC dvned evel Pure Mhs. / -D Co-Geomer i D is perpendiulr o he -is or prllel o he YZ-plne. ii B D is perpendiulr o he -is or prllel o he XZ-plne. iii C D is perpendiulr o he -is or prllel o he XY-plne. 9. Oher forms of equions of Plne B Three poin form The equion of he plne pssing hrough or onining hree non-olliner poins P,,, P,,, P,, is This is he hree-poin form of he plne. The inerep form If he -, - nd inerep of plne re,, respeivel, hen he equion of he plne is This is he inerep form of he plne. C Equion of plne hrough he poin he poin P,, nd wih [ B C] e he direion rios of he norml line of he plne. The equion of he plne is - B C. D Equion of he plne hrough or onining he line of inerseion of wo free plnes. e π B C D nd π B C D inerse he line. Then he plne pssing hrough he line n e epressed s B C D k B C D.* where k is onsn. N.B. The plne π nno e epressed in he form. Clss Eerises. Find he equion of he plne whih psses hrough he inerseion of he plnes π 5 nd π nd lso unis w from he origin.. plne is perpendiulr o eh of he plnes nd nd psses hrough he poin,,. Find he direion rios of he norml o he plne, he equion of he plne.. Relive Posiion eween line nd Plne Theorem. Given line whose direion rio re [ ] nd plneπ whose equion is B C D The line is // o he plne π if nd onl if B C Pge 7
HKCWCC dvned evel Pure Mhs. / -D Co-Geomer Pge 8 The line is o he plne π if nd onl if C B. N.B. The ngle eween line nd plne is defined s he ue ngle eween he line nd he orhogonl projeion of he line on he plne. e he direion rio of line e [ ] nd he equion of he plneπ is B C D. If θ is he ue ngle eween nd π, hen sin C B C B ϑ Clss Eerises Given plne π nd line......... 7 Trnsform he equion of in smmeri form nd find he ngle eween nd π. find he equion of he plne pssing hrough he poin P,, - nd eing prllel o he wo lines 5 nd Find he equion of he line pssing hrough he poin p, -, nd eing prllel o he wo plnes π ; nd π. Give our nswer in generl form. Find he equion of he plne onining he line 8 nd eing prllel o he line 7 5 Find he equion of he plne pssing hrough he poin P, -, nd eing perpendiulr o he line Find he equion of he plne onining he poin P,, - nd he line 7 Find he equion of he plne onining wo inerseing lines nd 8 Find he equion of he plne onining wo prllel lines nd 9 Find he lengh of he perpendiulr drwn from he poin P, 7, -9 o he line e π e plne wih norml veor n nd r e he posiion veor of n poin,, on he plne. Show h he equion of π in veor form is ρ n r where ρ is some rel onsn. e e line prllel o he veor nd pssing hrough he poin whose posiion veor is nd if r is he posiion veor of n poin,, on he line. Show h he equion of in veor form is r, where is rel prmeer. Theorem.
HKCWCC dvned evel Pure Mhs. / -D Co-Geomer Pge 9 Suppose h nd π inerse, show h he posiion veor of he poin of inerseion is n n ρ.. Condiion for Coplnr ines Two lines nd re oplnr if nd onl if Noe Sudens should dedue he ove heorem using previous knowledge. Clss Eerises Show h 5 nd re oplnr nd inerse. Tr o use differen mehods o solve i.. The Shores Disne eween Two Skew ines The shores disne eween wo skew lines nd re given Eerises E.. # 5 Theorem.