VIII - Geometric Vectors - SLC Home Page

MATHEMATICS 201-NYC-05Vectors and MatricesMartin HuardFall 2012VIII - Geometric Vectors1. Find all vectors in the following parallelepiped that are equivalent to the given vectors.E FABHGDCa) AB b) HE c) HAd) AB AEe) AB BCf) CD BFg) AD HEh) FA BGi) FG FEj) AB BC CE k) AB AD AEl) BC DC BDm) HC HA EC n) GH BE CE FA o) AG CB EC GA2. Let ABCDEF be a regular hexagon where AB a and FA b .a) Express the other sides, BC , CD , DE and EF , in terms of a and b .b) Express FB , FC and FD in terms of a and b .3. Let ABCDEFGH be a regular octagon where AB a and BC b .a) Express all other sides in terms of a and b .b) Express AC , AD , AE and DH in terms of a and b .4. Consider the vectors u , v and w such thatu 4, N15Ev 3, S45Ww 6, N60WFind the length and direction of the following vectors.a) u vb) u wc) wud) 1 12u 3ve) u w v f) 2 v3 w5

Math NYCVIII – Geometric Vectors5. An airplane has a maximum air speed of 500 km/h. If the plane is flying at is maximumspeed with a heading of 40 degrees west of north and the wind is blowing from north to southat 40 km/h, find the ground speed of the aircraft and the direction.6. A jet if flying through a wind that is blowing with a sped of 40 km/h in the direction N30 E.The jet has a speed of 610 km/h in still air and the pilot head the jet in the direction N45 E .Find the speed and direction of the jet.7. A woman launches a boat from on shore of a straight river and wants to land at the pointdirectly on the opposite shore. If the speed of the boat (in still water) Is 10 km/h and theriver is flowing east at the rate of 5 km/h, in what direction should she head the boat in orderto arrive at the desired landing point?8. A boat heads in the direction N72 E . The speed of the boat in still water is 24 km/h. Thewater is flowing directly south. It is observed that the true direction of the boat is directlyeast. Find the speed of the water and the true speed of the boat.9. Prove that if AB CD then AC BD .10. In the following parallelogram ABCD, M and N are the midpoints of AB and CDrespectively. Prove that AMCN is a parallelogram.BCMNAD11. Let ABCD be any quadrilateral such that OB OA OC OD where O is any point. Provethat ABCD must be a parallelogram.12. Prove that if the midpoints of the adjacent sides in a rectangle are joined, the resulting figureis a rhombus.13. Let ABCD be a parallelogram. Verify that the diagonal BD and the line AE, where E is themidpoint of BC, intersect at a point that divides both of these segments in the ratio of 2 to 1.14. Let ABCD be any quadrilateral on a plane. Prove that if P, Q, R and S are the midpoints ofsegments AB, BC, CD and AD respectively, then PQRS is a parallelogram.15. Prove that the line segment joining the midpoints of the diagonals in a trapezoid is parallel tothe base and is half the length of the difference between the lengths of the two bases.Fall 2012 Martin Huard 2

Math NYCVIII – Geometric Vectors16. Let ABCD be a parallelogram and let E divide the segment AB in a ratio of 2 to 1 and let Fdivide the segment DC in two. In what ratio does P, the point of intersection of AF and DE,divide the segments AF and DE?17. Determine whether the following statements are true of false. Explaina) Two equivalent vectors have the same initial point.b) 2AA 5AAc) If AB AC ADthen AB AC AD .d) 2AB and 3AB have the same direction.e) 2AB and 5BA have the same direction.f) If u kv then u v u v .g) There exists two nonzero vectors u and v such that u v u v .h) u u v vvi) 1vj) u and ku have the same direction.Fall 2012 Martin Huard 3

Math NYCVIII – Geometric VectorsAnswers1. a) EF DC HG b) GF DA CBc) GBd) AF DGe) AC EGf) CH BEg) 0 h) FH BDi) EG ACj) AE BF DH CG k) AG l) 0m) AE BF DH CG n) BG AHo) EB HC2. a) BC a b CD b DE a EF b ab) FB a b FC 2aBD 2a b3. a) CD 2b a DE b 2aEF a FG bGH a 2bHA 2a b AD 1 2b b) AC a bAE 2 2 b 2aDH 2b 2a4. a) u v 2.0, N32 W b) uw=8.0, N31 W c) wu=6.3, S82W1 1d)2u 3v 2.9, N25 E e) u w v 4.3, S73 E f) 2 v3 w53.5, S41E5. 470 km/h N43 W6. 648.7 km/h N44.1E7. N30 W8. 7.4 km/h and 22.8 km/h9. BD AD AC CDAD AB BDThus AC CD AB BDA10. We need to show that AM NC and AN MC .ForAC AB AB BDAC BDAM NC , AB DC since ABCD is a parallelogramAM MB DN NCAM AM NC NC2AM 2NCAM NCFor AN MC , we have MC MB BCC AM BCSince AB CDsince M and N are the midpoints of AB and CDsince M is the midpoint ofAB NC AD since AM NC and ABCD is a parallelogram DN AD AD DN ANsince N is the midpoint of CDFall 2012 Martin Huard 4

Math NYCVIII – Geometric Vectors11. We need to show that AB DC and BC AD .ForForAB DC , we have OB OA OC ODOB AO OC DOAO OB DO OCAB DCBC AD we have OB OA OC ODOD OA OC OBOD AO OC BOAO OD BO OCAD BC12. We need to show that PQRS is a parallelogram whose sides have equal length.BPAPQ PA AS SR RC CQ PA BQ SR AP QB AP QB SR AP QB SRQSSince ABCD is a rectangle and P,Q,R and S aremidpoints, thenAP PB DR RCAS SD BQ QCQR QP PS SR PQ PS SR SR PS SR PSTo show that all sides are of equal length, it suffices to show that PQproved that PQRS is a parallelogram.2 2 2PQ PB BQCRD(by the Pythagorean theorem) PS since we haveAP2 2ASPS2(by the Pythagorean theorem)hence PQPSFall 2012 Martin Huard 5

Math NYCVIII – Geometric Vectors2213. We need to show that if AP r AE then r and if DP sDB then s .ABrP3Since ABCD is a parallelogram and E is themidpoint of BC, then1 1BE BC AD2 2Let us express AP in terms of AB and BC in two different ways.AP r AEAP AD sDBr AB BEr AB r AB r2E1-s 1-rs12BCBC AD s DA AB 1 s AD s AB 1 s BC s AB AD s AD ABBy the basis theorem, we have the equations r s, r21 s .22Solving, we obtain r 3and s 3.14. We need to show that PQ SR and QR PS .DC3PBQCRADSFor PQ SRPQ PA AS SR RC CQ BP AS SR RC QB PB SD SR DR BQ PB BQ SD DR SR PQ SR SRThus 2PQ 2SRand PQ SR .For QR PS , we have QR QP PS SR PQ PS SR SR PS SR PSSince P and Q are the midpoints of AB and BCSince S and R are the midpoints of AD and CDBy associativityFall 2012 Martin Huard 6

Math NYCVIII – Geometric Vectors15. We need to show that MN 12AD BCBCWe have MN MC CB BNand also MN MA AD DN CM AD NB MC AD NBAdding, we obtain 2MN MC MC CB AD BN BNThus MN 12AD BCAM AD BCNDsince M and N are the midpoints of AC and DB16. If AP r AF and DP sDE , then we need to find r and s.Since ABCD is a parallelogram, we haveBCAB DC and AD BC .EPAlso, by the definitions of E and F, weFrhaves21ADAE 3AB and DF 2DC .Let us express AP in terms of AB andAD in two different ways.AP r AFr AD DFr AD r AD 12r2rDCABAP AD DPBy the basis theorem, we have the equationsr 1s43Solving, we obtain r 7and s 7divides DE in a ration of 3 to 4.r 2s2 3 AD sDE AD s DA AE AD s AD AB 2s1 s AD323AB. Hence, P divides AF in a ration of 4 to 3 andFall 2012 Martin Huard 7

Math NYCVIII – Geometric Vectors17. a) F. They may have any initial point as long as the have the same directions andmagnitude.b) T. AA 0 so 0 0.c) F. If AC , and AD have opposite direction thenDBCA D AAB AC AD .d) F. They have opposite direction.e) T. BA AB .f) F. If k 1 then u v and we have u v v v 0 0u v v v 1 v v 2 vg) T. If u 2vthen u v 2v v v and u v 2v v 2 v v vh) T. u u v u u v 1 v vi) T.vv1vv 1j) F. If k 1 then u and ku u have opposite direction.Fall 2012 Martin Huard 8