REVIEW SET A Using sle of m represents units, sketh vetor to represent: NON-CALCULATOR n eroplne tking off t n ngle of 8 ± to runw with speed of 6 ms displement of m in north-esterl diretion. Simplif: A C A + C DC. Construt vetor equtions for: If PQ, RQ p q r, nd RS, find SP. O p A [C] is prllel to [OA] nd is twie its length. Find, in terms of p nd q, vetor epressions for: q M AC OM. k j l n m C 6 Find m nd n if @ m A nd @ A re prllel vetors. n 7 If A! @ 7 A nd AC @ 6 A, find C. µ 8 If p, q, nd r, find: p ² q q ² (p r) 9 Consider points X(, ), Y(, ), W(, ), nd Z(, ). Use vetors to show tht WYZX is prllelogrm. Consider points A(, ), (, ), nd C(, k). Find k if AC is right ngle. Eplin wh: ² ² is meningless the epression ² does not need rkets. Find ll vetors whih re perpendiulr to the vetor.
In this question ou m not ssume n digonl properties of prllelogrms. OAC is prllelogrm with OA p nd OC q. M is the midpoint of [AC]. Find in terms of p nd q: i O ii OM Hene show tht O, M, nd re olliner, nd tht M is the midpoint of [O]. Find the vlues of k suh tht the following re unit vetors: O p A q 7 k M C k k Suppose j j, j j, nd j j. Find: ² ² ² 6 6 Find nd if J(,, ), K(,, ), nd L(,, ) re olliner. 7 Given j u j, j v j, nd u v @ A, find the possile vlues of u ² v. 8 [A] nd [CD] re dimeters of irle with entre O. If OC q nd O r, find: i D in terms of q nd r ii AC in terms of q nd r. Wht n e dedued out [D] nd [AC]? 9 Find t given tht @ t A nd @ t A re perpendiulr. t t + Show tht K(,, ), L(,, ), nd M(,, ) re verties of right ngled tringle.
REVIEW SET Cop the given vetors nd find geometrill: + CALCULATOR Show tht A(,, ), (,, ), nd C(,, ) re verties of n isoseles tringle. If r nd s find: j s j j r + s j j s r j Find slrs r nd s suh tht r + s. Given P(,, ) nd Q(,, ), find: PQ the distne etween P nd Q the midpoint of [PQ]. 6 If A(,, ), (,, ), C(,, ) re verties of tringle AC whih is right ngled t, find the vlue of. 7 Suppose @ A nd @ A. Find given. 8 Find the ngle etween the vetors i + j k nd i +j + k. 9 Find two points on the Z-is whih re 6 units from P(,, ). t Determine ll possile vlues of t if nd + t re perpendiulr. t Prove tht P( 6, 8, ), Q(, 6, 8), nd R(9,, 7) re olliner. If u @ A nd v @ A, find: u ² v the ngle etween u nd v. [AP] nd [Q] re ltitudes of tringle AC. Let OA p, O q, nd OC r. Find vetor epressions for AC nd C in terms of p, q, nd r. Using the propert ² ( ) ² ², dedue tht q ² r p ² q p ² r. Hene prove tht [OC] is perpendiulr to [A]. A Q C O P
Find two vetors of length units whih re prllel to i j + k. Find the mesure of D MC. D (,-, ) C (,-, ) 6 Find ll vetors perpendiulr to oth 7 Find k given tht @ k p k A (-,, -) @ A nd @ A. C A is unit vetor. M (,, -) Find the vetor whih is units long nd hs the opposite diretion to 8 Find the ngle etween @ A nd @ A. @ A. 9 Determine the mesure of Q DM given tht M is the midpoint of [PS]. Q P M R S 7 m A m C D m REVIEW SET C Find single vetor whih is equl to: PR + RQ PS + SQ + QR For m @ 6 A, n @ A, nd p @ A, find: 6 m n + p n p j m + p j Wht geometril fts n e dedued from the equtions: A CD A AC? Given P(,, 6) nd Q(, 7, 9), find: the position vetor of Q reltive to P the distne from P to Q the distne from P to the X-is.
P Q In the figure longside, OP p, OR r, nd M RQ q. M nd N re the midpoints of [PQ] nd p O N q [QR] respetivel. Find, in terms of p, q, nd r: OQ PQ ON d MN r R 6 Suppose p @ A, q @ A, nd r @ A. Find if: p q r 7 Suppose j v j nd j w j. If v is prllel to w, wht vlues might v ² w tke? 8 Find unit vetor whih is prllel to i + rj +k nd perpendiulr to i +j k. 9 Find t if @ t +A nd @ t +ta re perpendiulr vetors. t Find ll ngles of the tringle with verties K(,, ), L(,, ), nd M(,, ). µ Find k if the following re unit vetors: @ k k A k k Use vetor methods to find the mesure of G AC in the retngulr o longside. F E G H m Using p, q, nd r, verif tht: p ² (q r) p ² q p ² r. P(,, ) nd Q(,, ) re two points in spe. Find: PQ the ngle tht PQ mkes with the X-is. MP, Suppose OM, Write down the two possile position vetors OT. MP ² PT, nd j MP j j PT j. 6 Given p i j +k nd q i j +k, find the ngle etween p nd q. 7 Suppose u i + j, v j, nd µ is the ute ngle etween u nd v. Find the et vlue of sin µ. 8 Find two vetors of length units whih re perpendiulr to oth i +k nd i j + k. A D 8 m C m
REVIEW SET A NON-CALCULATOR For the line tht psses through ( 6, ) with diretion, write down the orresponding: vetor eqution prmetri equtions Crtesin eqution. 8 7 (, m) lies on the line with vetor eqution + t. Find m. Line L hs eqution r + t. Lote the point on the line orresponding to t. Eplin wh the diretion of the line ould lso e desried. Use our nswers to nd to write n lterntive vetor eqution for line L. P(,, ), Q(,, ), nd R(,, ) re three points in spe. Find prmetri equtions of line (PQ). Show tht if µ P QR, then os µ p p. 6 Tringle AC is formed three lines: µ Line (A) is + t µ Line (AC) is + u. Line (C) is. s, t, nd u re slrs. Use vetor methods to find the oordintes of A,, nd C. Find j A j, j C j, nd j AC j. Clssif tringle AC. 7 + s. 6 Consider two unit vetors nd. Prove tht the vetor + isets the ngle etween vetor nd vetor. Consider the points H(9,, ), J(7,, ), nd K(,, ). Find the eqution of the line L tht psses through J nd isets H JK. Find the oordintes of the point where L meets (HK). 7 Suppose A is (,, ) nd is (,, ). Write down vetor eqution of the line through A nd. Find the eqution of the plne through with norml A!. Find two points on (A) whih re p units from A. 8 For C(,, ) nd D(,, ), find the oordintes of the point where the line pssing through C nd D meets the plne with eqution + z.
9 Suppose OA, O, j j, j j p 7, nd i +j k. Find: ² the re of tringle OA. How fr is X(,, ) from the plne z 8? Find the oordintes of the foot of the perpendiulr from Q(,, ) to the line z. Find if possile the point where the line through L(,, ) nd M(,, ) meets the plne with eqution z. Find the shortest distne from L to the plne. The equtions of two lines re: L : t, t +, z t L : z. Find the point of intersetion of L nd the plne + z. Find the point of intersetion of L nd L. Find the eqution of the plne tht ontins L nd L. Show tht the line + find the distne etween them. z is prllel to the plne 6 +7 z 8 nd + + z 6 is the eqution of sphere with entre (,, ) nd rdius p 6 units. Find the point(s) where the line through (,, ) nd (,, ) meets the sphere. When n rher fires n rrow, he is suddenl wre of reeze whih pushes his shot off-trget. The speed of the shot j v j is not ffeted the wind, ut the rrow s flight is ± off-line. Drw vetor digrm to represent the sitution. Hene eplin wh: i the reeze must e 9 ± to the intended diretion of the rrow ii the speed of the reeze must e j v j sin ±. 6 In the figure ACD is prllelogrm. X is (,,-) X the midpoint of C, nd Y is on [AX] suh tht AY! YX!. Y Find the oordintes of X nd D. Find the oordintes of Y. Show tht, Y, nd D re olliner. A (,,-) D 8 < + z 7 Solve the sstem + z : 7 + + kz k for n rel numer k using row opertions. Give geometri interprettions of our results. C (,,)
REVIEW SET CALCULATOR Find the vetor eqution of the line whih uts the -is t (, 8) nd hs diretion i +j. A ht is siling with onstnt speed p km h in the diretion i j. Initill it is t point ( 6, ). A eon is t (, ) t the entre of tin toll. Distnes re in kilometres. Find in terms of i nd j : i the initil position vetor of the ht ii the diretion vetor of the ht iii the position vetor of the ht t n time t hours, t >. Find the time when the ht is losest to the eon. If there is reef of rdius 8 km round the toll, will the ht hit the reef? Write down i vetor eqution ii prmetri equtions for the line pssing through: (, ) with diretion (, 6, ) nd (,, ). A smll plne n fl t km h in still onditions. Its pilot needs to fl due north, ut needs to del with 7 km h wind from the est. In wht diretion should the pilot fe the plne in order tht his resultnt veloit is due north? Wht will the speed of the plne e? Find the ngle etween line L pssing through (, ) nd (, ), nd line L pssing through (, ) nd ( 6, 7). 6 Sumrine X is t (, ). It fires torpedo with veloit vetor t etl :7 pm. Sumrine Y8 is t (, ). It fires torpedo with veloit vetor t :9 pm to interept the torpedo from X. Distne units re kilometres. t is in minutes. Find (t) nd (t) for the torpedo fired from sumrine X. Find (t) nd (t) for the torpedo fired from sumrine Y8. d At wht time does the intereption our? Wht ws the diretion nd speed of the intereption torpedo? 7 Suppose P is the plne z 9 nd P is the plne + +z. L is the line with prmetri equtions t, t, z t. Find the ute ngle etween: L nd P P nd P.
8 Consider the lines L : z t. 8 +9 6 z 7 nd L : +t, 9+8t, Show tht the lines re skew. Find the ute ngle etween them. d Line L is trnsltion of L whih intersets L. Find the eqution of the plne ontining L nd L. Find the shortest distne etween them. 9 Find the eqution of the plne through A(,, ), (,, ), nd C(,, ). If X is (,, ), find the ngle tht (AX) mkes with this plne. Find ll vetors of length units whih re norml to the plne + z 6. Find unit vetor prllel to i + rj +k nd perpendiulr to i j +k. The distne from A(,, ) to the plne with eqution +z k is units. Find k. Find the ngle etween the lines with equtions nd + 7. Consider A(,, ) nd (,, ). Find the vetor eqution of the line through A nd. Hene find the oordintes of C on (A) whih is units from A. Find the ngle etween the plne + z nd the line t, t +, z t +. Let r i j k, s i + j +k, t i +j k, e the position vetors of the points R, S, nd T, respetivel. Find the re of the tringle RST. Clssif the following line pirs s either prllel, interseting, or skew. In eh se find the mesure of the ute ngle etween them. +t, +t, z t nd 8+s, s, z 7 s +t, t, z +t nd s, +s, z +s 6 p @ A nd q @ A Find p q. Find m if p q is perpendiulr to the line L with eqution r @ A + @ A. m Hene find the eqution of the plne P ontining L whih is perpendiulr to p q. d Find t if the point A(, t, ) lies on the plne P. e For the vlue of t found in d, if is the point (6,, ), find the et vlue of the sine of the ngle etween (A) nd the plne P. 7 Show tht the plne + + z ontins the line L : t +, t, z t +, t R. For wht vlues of k does the plne + k + z ontin L? Hene find the vlues of p nd q for whih the following sstem of equtions hs n infinite numer of solutions. Clerl eplin our resoning. 8 < + + z + z : + p +z q
8 < +z 8 Consider the sstem + +( k)z : +6 + kz where k n tke n rel vlue. Redue the sstem to ehelon form. For wht vlue of k does the sstem hve no solutions? Interpret this result geometrill. i For wht vlue(s) of k does the sstem hve unique solution? ii Find the unique solution in terms of k, nd interpret the result geometrill. iii Find the unique solution when k. REVIEW SET C Find the veloit vetor of n ojet moving in the diretion i j with speed km h. A moving prtile hs oordintes P((t), (t)) where (t) +8t nd (t) +6t. The distne units re metres, nd t > is the time in seonds. Find the: initil position of the prtile position of the prtile fter seonds prtile s veloit vetor d speed of the prtile. Trpezium KLMN is formed the following lines: (KL) is + p. (ML) is + q. 9 6 (NK) is + r. (MN) is + s. 7 9 p, q, r, nd s re slrs. Whih two lines re prllel? Eplin our nswer. Whih lines re perpendiulr? Eplin our nswer. Use vetor methods to find the oordintes of K, L, M, nd N. d Clulte the re of trpezium KLMN. Find the ngle etween the lines: L : t, t nd L : +s, s. Consider A(,, ) nd (,, ). Find j A j. Show tht the line pssing through A nd n e desried r j k + ( i + j k) where is slr. Find the ngle etween (A) nd the line with vetor eqution t(i + j + k).
6 Let i represent displement km due est nd j represent displement km due north. Rod A psses through ( 9, ) nd (, 6). Rod psses through (6, 8) nd (, 8). Find vetor eqution for eh of the rods. An injured hiker is t (, ), nd needs to trvel the shortest possile distne to rod. Towrds whih rod should he hed, nd how fr will he need to wlk to reh this rod? - (-9, ) - - - H(, ) (, 8) (,-6) ( 6,-8) A 7 Given the points A(,, ), (,, ), nd C(9,, ): Show tht A is perpendiulr to AC. Find the eqution of the line through: i A nd ii A nd C. 8 The tringle with verties P(,, ), Q(,, ), nd R(,, ) hs re p 8 units. Find. 9 Consider A(,, ), (,, ), nd C(,, ). Find the eqution of the plne defined A,, nd C. Find the mesure of C A. D(r,, r) is point suh tht DC is right ngle. Find r. Given A(,, ), (,, ), nd C(,, ), find: d the norml vetor to the plne ontining A,, nd C D, the fourth verte of prllelogrm ACD the re of prllelogrm ACD the oordintes of the foot of the perpendiulr from C to the line A. P(,, ), Q(,, ), nd R(,, ) re three points in spe. Find: PQ, j PQ j, nd QR the prmetri equtions of (PQ) vetor eqution of the plne PQR. Given the point A(,, ), the plne +z 8, nd the line defined 7 t, 6+t, z +t, find: the distne from A to the plne the oordintes of the point on the plne nerest to A the shortest distne from A to the line. Find the eqution of the plne through A(,, ), (,, ), nd C(,, ). Find the eqution of the line, in prmetri form, whih psses through the origin nd is norml to the plne in. Find the point where the line in intersets the plne in.
Consider the lines with equtions +t, +t, z t. z + nd Are the lines prllel, interseting, or skew? Justif our nswer. Determine the ute ngle etween the lines. 8 Line hs eqution +9 6 z. 7 Line hs vetor eqution @ A @ 9 A + @ 8 A. z Show tht lines nd re skew. Line is trnsltion of line whih intersets line. Find the eqution of the plne ontining lines nd. Hene find the shortest distne etween lines nd. d Find the oordintes of the points where the ommon perpendiulr meets the lines nd. 6 Lines L nd L re defined L : r @ A + s @ A nd L : r @ A + t @ A. Find the oordintes of A, the point of intersetion of the lines. Show tht the point (,, ) lies on the line L. Find the eqution of the line C given tht C(,, ) lies on L. d Find the eqution of the plne ontining A,, nd C. e Find the re of tringle AC. f Show tht the point D(9,, ) lies on the norml to the plne pssing through C. 7 Three plnes hve the equtions given elow: Plne A: + +z Plne : + +9z Plne C: +6z 8 Show tht plne A nd plne interset in line L. Show tht plne nd plne C interset in line L. Show tht plne A nd plne C interset in line L. d Show tht L, L, nd L re prllel ut not oinident. e Wht does this men geometrill?
ANSWERS 9 REVIEW SET A 6 ms - 8 Sle: m ms - REVIEW SET + - - - AC AD N m Sle: m m q p + r l k j + n m p + q p + q 6 m, n 8 7 8 7 8 6 k 6 ³ ² is slr, so ² ² is slr dotted with vetor, whih is meningless. must e done first otherwise we hve the ross produt of slr with vetor, whih is meningless. t ³, t 6 i p + q ii p + q k p 7 k p ² ² ² 6, 7 If µ is ute, u ² v p 99; if µ is otuse, u ² v p 99. 8 i q + r ii r + q D AC, [D] k [AC] 9 t LM! ) M 9 ±, KM! A AC p units nd C p 6 units ) is isoseles p units p units p 9 units r, s 7 @ 6 p 6 units (,, ) 6 7 A 8 6: ± 9 (,, ) nd (,, 9) t or 8 ¼ 6:± AC p + r, C q + r p (i j + k) ¼ 6: ± 6 k 7 k p 8 ¼ 8: ± 9 ¼ 6: ± REVIEW SET C PQ PR 7 6 p 7 units A CD, [A] k [CD] C is the midpoint of [A]. PQ p 6 units p 6 units r + q p + r + q r + q d p + r
9 ANSWERS 6 8 @ A p i + p k 9 t p K ¼ :7 ±, L ¼ : ±, M ± 7 v ² w 6 k k p ¼ :7 ± PQ ³ OT 8 or ³ ¼ :8 ± 7 sin µ p 8 p 7 9 6 ¼ 6:6 ± REVIEW SET A ³ ³ ³ 6 + t, t R 6+t, t, t R + 6 m (, ) ³ ³ ³ + s is non-zero slr multiple of ³ +t, t, z t, t R j PQ ² QR j Use os µ j PQ! jj QR! j A(, ), (6, ), C(8, ) ³ j A! j p units, j C! j p 8 units, j AC! j p units isoseles 6 OAC is rhomus. So, its digonls iset its ngles. 7 z (7,, ) z 7 +z + O @ + A A, R +, R (,, 9) 8 (6,, ) 9 7 or (,, ) p units 7 units ( 8, 7, ) The do not meet, the line is prllel to the plne. p 6 units (, 7, 9 ) (,, ) 6 8 z C p units (,, ) nd (,, ) trget reeze i isoseles tringle ) remining ngles 89 ± eh, reeze mkes ngle of 8 89 9 ± to intended diretion of the rrow. ii iset ngle ± nd use sin ± speed jvj ) speed jvjsin ± 6 X(7,, ), D(7,, ) Y(,, ) D! nd Y! So, D!! Y, et. 7 If k, the plnes meet in the line, + t, z t, t R. If k 6 the plnes meet t the point (,, ). REVIEW SET ³ ³ 8 + ³, R i 6i +j ii i j iii ( 6 t)i + ( t)j, t > t :8 h shortest distne ¼ 8:8 km, so will miss reef ³ ³ ³ i + t, t R ii +t, t, t R 6 i 6 + t 8, t R z ii +6t, 6 8t, z t, t R : ± est of due north ¼ km h 8: ± 6 X, +t, t, t > Y8, t, + t, t > intereption ourred t :: pm d ering ¼ ± west of south, ¼ : units per minute 7 ¼ :8 ± ¼ 6:9 ± 8 ¼ 8:6 ± + +6z 7 d units 9 + + z ¼ 6: ± p p p p p nd p p 7 i + p8 7 j + p 7 k or p 7 i p8 7 j p 7 k k 7 or 7: ± or 7:6 ± z v intended diretion v tul flight +, R
98 ANSWERS ³ p, + p, p 6 6 6 ³ + p, p, + p 6 6 6 7:8 ± 9p units interseting t (,, ), ngle ¼ : ± skew, ngle ¼ 7: ± 6 m z d t e p 7 k t(p + ) q + hs infinitel mn solutions for t when p + nd q +, ) p, q 8 k k + nd No solutions if k. Two plnes re prllel nd interseted the third plne. i Unique solution when k 6. ii +,, z iii (,, ) k + k + Plnes meet t point. REVIEW SET C p (i j) (, ) (8, 7) (KL) is prllel to (MN) s ³ 8 6 ³ (KL) is perpendiulr to (NK) s nd (NK) is perpendiulr to (MN) s d ms ³ is prllel to ³ ² ³ ² ³ ³ K(7, 7), L(, ), M(, ), N(, 7) d 6 units : ± j A j p 7 units A lies on the line r where nd lies on r where ) the line etween A nd is the sme s line r, so it n e desried r. 7: ± 6 Rod A: Rod : ³ ³ Rod, km 7 A ² AC ³ 9 ³ 6 8 6 ² ³ + + ¹, R ³, ¹ R i t, t, z +6t, t R ii +s, +s, z +s, s R 8 or 6 9 +9 z ¼ :9 ± r p n D(,, ) ¼ :8 units d ( 6, 6, ) PQ, j PQ j p 6 units, +,, z, R z + + ¹ QR units (,, ) p 6 units,, ¹ R + +z t, t, z t, t R (,, 7 ) interseting t (,, ) µ ¼ 7: ± + +6z 7 units d (, 7, ) on line nd (9,, ) on line 6 A(,, ) r + u d +z 7 e p units f norml is z + 7 e The three plnes hve no ommon point of intersetion. The line of intersetion of n two plnes is prllel to the third plne.