MCVU Thursda Ma, Review of Equatios of a Straight Lie (-D) U8L Sec. 8.9. Equatios of Lies i R Cosider the lie passig through A (-,) with slope, as show i the diagram below. I poit slope form, the equatio of the lie is If we solve this equatio for, we obtai the slope & -itercept form If we rearrage the terms, we obtai the equatio i stadard form The equatio of a lie is satisfied b the coordiates of all poits o the lie ad o others. For eample, the poit (5,8) is o the lie above. These coordiates satisf all three equatios. I the et sectio, we will be cosiderig the equatios of lies i R. Noe of these previous forms for the equatios of lies ca be eteded to lies i three dimesios. The above equatios all ivolve slope which caot be eteded to R because slope ivolves ol two quatities. The slope of a lie i R is ot defied. We are ow goig to establish other forms of the equatio of a lie i R that ca be easil eteded to R. These forms ivolve vectors. The diagram above goes through the poit A(-,) ad has a directio vector m,. See diagram below. Directio vectors of lies are ot uique. A scalar multiple of [,] is also a directio vector of this lie. Page of Prepared b Mr. S. McEwe
MCVU Thursda Ma, Vector Equatio of a Lie i R Let P (,) be a poit o the lie. Visualie P movig back ad forth alog the lie. As it moves, poits O, A, ad P alwas form a triagle i which the triagle law is satisfied Sice AP is colliear with m,, we kow that, where t is a scalar. Let OA a ad OP p. The we ca write the above equatio as This is called the vector equatio of a lie. To determie other poits o the lie we substitute differet scalars for t. If t = the we ad if t = - we get. Parametric Equatio of a Lie i R It is more efficiet to rewrite the vector equatio so that the right side is a sigle vector before substitutig the values for our parameter t. These are called the parametric equatios of a lie. Parametric equatios of a lie have the followig properties Eample # A lie passes through the poits A (-,) ad B (5,). a) Write a vector equatio for the lie. b) Write a parametric equatio for the lie. Solutio Page of Prepared b Mr. S. McEwe
MCVU Thursda Ma, Smmetric Equatio of a Lie i R Suppose a lie passes through the poit A (,-) ad has a directio vector m, are = + t = - + t If we solve each equatio for t, we obtai. It s parametric equatios Sice the values of parameter t must be the same i each equatio, these two epressios are equal. Hece This is called the smmetric equatio of a lie. Smmetric equatios have the followig properties Eample # 6 The smmetric equatio of a lie is. a) Write the parametric equatios of the lie. b) Determie the coordiates of aother poit o the lie. Solutio Vector, parametric ad smmetric equatios of a lie i R represet a differet wa of thikig about the equatio of a lie. Whe we use parametric equatios for two differet lies i the same problem, we eed to use differet letters for the parameters of each lie. Page of Prepared b Mr. S. McEwe
MCVU Thursda Ma, Eample # Smmetric equatios of two lies are give as L 5 ad L Fid the coordiates of the poit of itersectio of L ad. Solutio L Homework Hadout 8.9. Page of Prepared b Mr. S. McEwe
MCVU Thursda Ma, U8L Sec. 8.9. Equatios of Lies i R The three ew forms of lies i R ca ow be eteded to R. Let A (a, a, a ) be a fied poit o a lie i R with directio vector m m, m m poit o the lie. The equatios of the lie ca be writte i the followig forms; Vector Equatio Parametric Equatio,. Let P (,,) be a Smmetric Equatio Eample # Write vector, parametric ad smmetric equatios for the lie through the poits A (5,,-) ad B (,5,-). Solutio Page 5 of Prepared b Mr. S. McEwe
MCVU Thursda Ma, I three dimesios, there are three itersectio possibilities for two distict lies. Eample # Fid the coordiates of the poit of itersectio for the followig lies; Solutio L ; ad L Homework Hadout 8.9. Page 6 of Prepared b Mr. S. McEwe
MCVU Thursda Ma, U8L Sec. 8.9. Equatios of Plaes A plae is determied b. For eample, the diagram below shows the plae passig through the poit A (-,5,) ad cotais the vectors u,, ad v,, tiltig upwards awa from the viewer, as idicated b the triagle formed b the vectors u tails are at the origi.. The plae is ad v whe their Vector Equatio of a Plae Let P (,,) be a poit o this plae. Visualie P movig aroud the plae i a positio. As it moves, poits O, A ad P alwas form a triagle i which the triagle law is satisfied the plae, we kow that AP is a liear combiatio of u ad scalars. Therefore, we ca write the above equatio as v. Hece, OP OA AP Sice P is o AP s u t v where s ad t are a This is called the vector equatio for the above plae. Parametric Equatio of a Plae As doe previousl for lies i R ad R, we ca write the vector equatio of the plae i a differet wa; Therefore, This is called the parametric equatio for the above plae. Page 7 of Prepared b Mr. S. McEwe
MCVU Thursda Ma, Eample # Write vector ad parametric equatios for the plae which cotais the poits A (,,-), B (5,,) ad C (,,-6). Solutio Cartesia Equatio of a Plae (also kow as Scalar Equatio of a Plae) This is a equatio without usig parameters. To determie the scalar equatio of a plae, we eed to fid the ormal vector for the plae (ie a vector perpedicular to the plae). This ca be doe b takig the cross product of two o-colliear vectors i the plae. The result is. Eample # Determie the Scalar Equatio of the Plae usig the same poits from eample #. Solutio Homework Hadout 8.9. Page 8 of Prepared b Mr. S. McEwe
MCVU Thursda Ma, U8L Sec. 8.9. Problems Ivolvig Lies & Plaes I three dimesios, there are three itersectio possibilities for a lie ad a plae. Eample # Determie the poit(s) of itersectio of each lie ad plae, if possible. a) L [,,] = [,,6] + t [,-,] ad 7 6 b) L ad 8 c) L = 5 + t ad 5 = - + t = 9 t Solutio Homework Hadout 8.9. Page 9 of Prepared b Mr. S. McEwe
MCVU Thursda Ma, U8L5 Sec. 8.9.5 Parallel or Itersectig Plaes & Skew Lies Skew lies are lies i three dimesios that are ot parallel ad do ot itersect. Two skew lies ma lie i parallel plaes. To show that lies are skew lies; Eample # Show that L ad L are skew lies. L ad L Solutio Page of Prepared b Mr. S. McEwe
MCVU Thursda Ma, Page of Prepared b Mr. S. McEwe Fidig Parallel Plaes Eample # Determie the equatios of two parallel plaes that cotai the skew lies L ad L Solutio Two distict plaes ma be either parallel or itersectig. It is eas to distiguish these two cases because parallel plaes have colliear ormal vectors. For eample, the followig plaes are parallel because their ormal vectors [,-,] ad [6,-,8] respectivel, are colliear. Ie 9 8 6 7 The followig plaes are ot parallel, thus the itersect i a lie. Ie 5 7 These two equatios, take together, ca be regarded as equatios of the lie. However, the are ot ver useful i this form because the do ot cotai specific iformatio about the lie such as a directio vector or the coordiates of a poit o the lie.
MCVU Thursda Ma, Eample # Fid parametric ad smmetric equatios of the lie of itersectio of the plaes 7 ad 5 Solutio Homework Hadout 8.9.5 Page of Prepared b Mr. S. McEwe
MCVU Thursda Ma, U8L6 Sec. 8.9.6 Two Itersectig Plaes Cotiued. I ma problems ivolvig the lie of itersectio of two plaes, it is ot ecessar to determie the equatio of the lie. Sometimes ol a directio vector of the lie is eeded. The followig diagram shows that the directio vector of the lie of itersectio of two plaes is perpedicular to the ormal vectors of each plae. Eample # Fid the equatio of a ew plae that passes through the poit A (,-,) ad is perpedicular to the lie of itersectio of the plaes ad 7 Solutio Liear Combiatios of Equatios of Plaes Ifiitel ma plaes pass through a give lie i space. I problems ivolvig a plae passig through the lie of itersectio of two plaes, the followig approach is ver effective. Cosider the plaes ad with these equatios; We ca tell b ispectio that the ormal vectors of ad are. Hece, these two plaes have a. Suppose we combie their equatios as follows. Multipl Multipl b a scalar, s s b a scalar, t t Page of Prepared b Mr. S. McEwe
MCVU Thursda Ma, A ew plae ca be created b addig these two; s t This is a liear combiatio of the previous two plaes. Assumig that s, we ca divide both sides b s to t obtai s t Sice t ad s are both real umbers, their quotiet is also a real umber. Hece, we ca replace with a sigle s smbol, k. The the equatio becomes k This equatio is sigificat because a poit o the lie of itersectio of plaes ad also lies o. Eample # Fid the equatio of the plae passig through the lie of itersectio of the plaes ad, ad satisfies the give coditios a) The plae passes through he poit A (,,) b) The plae is also parallel to the plae 5 7 6 Solutio Page of Homework Hadout 8.9.6 Prepared b Mr. S. McEwe
MCVU Thursda Ma, U8L7 Sec. 8.9.7 Problems Ivolvig Three Plaes Da # Three plaes ca iteract i we will look at Tpe IV V.. Toda we will be lookig at Tpe I III ad tomorrow Tpe I Three Parallel Plaes Whe plaes are parallel, the ormals are The equatios below represet this situatio. We ca tell this because all three ormal vectors [,-,], [9,-6,] ad [6,-,] are colliear. That is, ad. The plaes are because their equatios do ot satisf these relatioships. The plaes are distict, separated because their costat term does ot follow the same scalar multiples as the ormals. ie 9 6 6 5,, 9, 6, 6,, Tpe II Two Parallel Plaes & Third Plae Itersects Both Suppose we replace plae with aother plae,, that is ot parallel to the other two plaes. The itersects ad formig two parallel lies. The side view shows the plaes ad as two parallel lies with a third lie,, itersectig them. The ormal vectors lie flat o the page ad are coplaar. Two of the ormals are still. but ot with the rd. Plaes are still distict. Normals are Page 5 of Prepared b Mr. S. McEwe
MCVU Thursda Ma, Page 6 of Prepared b Mr. S. McEwe ie,, 7 6, 9, 6 9,, Tpe III Plaes Itersect i Pairs Suppose we replace plae with aother plae, 5, that is ot parallel to either of the other two plaes. The plaes, 5, ad itersect i pairs, formig three parallel lies. The side view shows the plaes as lies formig a triagle. Agai, the ormal vectors lie flat o the page ad are coplaar. The ormal vector of oe plae is a of the other ormals, but the equatios do t follow the same liear combiatios. The three ormals are still coplaar. ie 5,, 5 5,, 7,, 5 5 I all three tpes, each sstem of equatios has because there is o poit o all three plaes. It is impossible for the coordiates of a poit to satisf all three equatios. We sa that each sstem of equatios is. Eample # Describe how the plaes i each liear sstem are related. If there is a uique poit or lie of itersectio, determie its coordiates or equatio. a) 5
MCVU Thursda Ma, Page 7 of Prepared b Mr. S. McEwe Solutio b) 8 5 5 Solutio Homework Hadout 8.9.7
MCVU Thursda Ma, Page 8 of Prepared b Mr. S. McEwe U8L8 Sec. 8.9.8 Problems Ivolvig Three Plaes Da # Tpe IV Three Plaes Itersect i a Lie Oe ormal vector is a of the other ormals ad the also satisf this same liear combiatio. All ormals are still coplaar. ie 5,8 5, 6 8 5 5,,,, 6 6 The ormal vector of 6, [5,-5,8], is a liear combiatio of the other two ormal vectors. The plaes itersect i a sigle lie because the equatios satisf this same liear combiatio. That is Eample # Show that the followig three plaes itersect at a lie ad fid the parametric equatios of that lie of itersectio.,,,, 7,, Solutio
MCVU Thursda Ma, Tpe V Three Plaes Itersect at a Sigle Poit This is the ol situatio out of the 5 tpes where the ormal vectors are epress oe of the ormal vectors as a liear combiatio of the other two ormals.. It is ot possible to To determie if there is just oe poit of itersectio, use the check for coplaar vectors. If the the ormals are ot coplaar ad there is a sigle poit of itersectio. For Tpe IV & V ol, the set of equatios for plaes because all plaes have a commo itersectio. These sstems of equatios are called. Eample # Show that the followig plaes itersect at a sigle poit ad the determie the coordiates of the poit of itersectio. Solutio 5 7,,,,,, Page 9 of Homework Hadout 8.9.8 Prepared b Mr. S. McEwe
MCVU Thursda Ma, U8L9 Sec. 8.9.9 Usig Matrices to Solve Sstems of Equatios We ofte ecouter sstems of equatios i which the cotet is ot lies ad plaes. Ecoomists ofte have to work with sstems of does of equatios ivolvig does of ukows. Relig o a geometric iterpretatio would ot be ver useful. To help orgaie such vast amouts of data, mathematicias have created a powerful tool called a. A matri is simpl umerical data arraged i a rectagular arra. We usuall eclose the arra i square brackets. The method of solvig a liear sstem amouts to combiig liear combiatios of the equatios i certai was. Sice several similar steps are ivolved, this method is ideall suited for techolog. Solvig a liear sstem usig techolog requires a sstematic approach because a calculator or computer uses the same method ever time. Elemetar Row Operatios A sstem of liear equatios ca be represeted b a matri. To obtai a equivalet sstem, perform a of these operatios. A sstem of three liear equatios i,, ad represets three plaes i R. It ca be represeted b a matri * * * * havig the form; * * * *, called the where each * represets a real umber. * * * * Whe we solve the sstem usig matrices, we attempt to use the to obtai a # matri havig the form # called the. The method of doig this is called #. The matri that results is the reduced matri ad is i Eample # (Solvig a Matri i R ) Solve the liear sstem usig matrices. 6 Solutio 8 6 Page of Prepared b Mr. S. McEwe
MCVU Thursda Ma, Eample # (Solvig a Matri i R ) Solve the liear sstem usig matrices 9 8 Solutio Homework Hadout 8.9.9 U8L Sec. 8.9. Matrices Cotiued * * * * # It is ot alwas possible to reduce a matri of the form * * * * to oe of the form # usig * * * * # row reductio. If oe of the equatios is a liear combiatio of the other two, a will * # occur at some poit. It ma be possible to reduce the matri to a form such as * #. The the sstem is, ad the correspodig plaes itersect i a lie. Parametric equatios of this lie costitute the solutio of the sstem. Page of Prepared b Mr. S. McEwe
MCVU Thursda Ma, Page of Prepared b Mr. S. McEwe Eample # Solve the sstem usig matrices ad iterpret the solutio geometricall. 9 6 7 Solutio
MCVU Thursda Ma, Eample # Solve the followig sstem of liear equatios b reducig the correspodig augmeted matri to row-reduced echelo form. w Solutio w w 8 w Hadout (Represetatios of Solutios to Sstems of Equatios i R ) Homework Hadout 8.9. Page of Prepared b Mr. S. McEwe