April 8, 2017 Math 9. Geometry. Solving vector problems. Problem. Prove that if vectors and satisfy, then.
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1 pril 8, 2017 Mth 9 Geometry Solving vetor prolems Prolem Prove tht if vetors nd stisfy, then Solution 1 onsider the vetor ddition prllelogrm shown in the Figure Sine its digonls hve equl length,, the prllelogrm is retngle (this is euse the digonls divide it into pirs of ongruent isoseles tringles) D + - Solution 2 Prolem Show tht for ny two non-olliner vetors nd in the plne nd ny third vetor in the plne, there exist one nd only one pir of rel numers suh tht n e represented s Solution Let us drw prllelogrm, whose digonl is the segment,, nd the sides nd re prllel to the vetors nd, respetively Sine, there exists numer x, suh tht Similrly, there exists numer y, suh tht Then, Prolem Derive the formul for the slr (dot) produt of the two vetors, nd,, using their representtion vi two perpendiulr vetors of unit length, nd, direted long the nd the xis, respetively e y Y e x (, ) x y (, ) x y X Solution It is ler from the Figure tht
2 Prolem Vetors, nd re represented y the rdil segments direted from the entre of the irle to points, nd on the irle (see Figure) Wht re the ngles, nd, if Solution It is ler from the Figure tht if, then nd In the seond se the sitution looks similr to tht in the figure, ut with points nd interhnged Therefore, nd Prolem Vetors, nd re represented y the internl isetors in the tringle, direted from eh vertex to the point on the opposite side (see figure) Express the sum, through vetors nd (nd the sides of the tringle,,, ) For wht tringles does this sum equl 0? Prolem Given three vetors,, is perpendiulr to nd, show tht vetor Solution Let us find the slr produt, Whih mens tht is perpendiulr to Prolem Given tringle, find the lous of points suh tht Using this finding, prove tht three ltitudes of
3 the tringle re onurrent (ie ll three interset t ommon rossing point, the orthoenter of the tringle ) Solution Let e n ritrry point on the plne Express (see Figure),, Then, oviously, M Hene, ll points on the plne stisfy the given vetor ondition Now, let e the rossing point of the two ltitudes of the tringle, nd Then, y the definition of n ltitude However, we hve just proved tht for ny point, H inluded, n ltitude Therefore,, nd is lso Prolem Let e the irumenter ( enter of the irle irumsried round) nd e the orthoenter (the intersetion point of the three ltitudes) of tringle Prove tht, Solution Let, nd, e the dimeters of the irumirle of the tringle Then, qudrilterls nd re retngles (they re mde of pirs of insried right tringles whose hypotenuse re the orresponding dimeters), nd H1 is prllelogrm Therefore, nd Now, 1 H 1 1
4 Vetor pproh to the rhimedes method of enter of mss Let us ssume tht system of geometri points, hs msses ssoited with eh point The totl mss of the system is y definition, the enter of mss of suh system is point, suh tht For the se of just two mssive points, { } nd { } this redues to, the rhimedes fmous lever rule Heuristi properties of the enter of Mss 1 Every system of finite numer of point msses hs unique enter of mss (M) 2 For two point msses, nd, the M elongs to the segment onneting these points; its position is determined y the r h mede lever rule: the p t m t me the distne from it to the M is the sme for oth points, 3 The posit f the y tem e ter f m d e not hnge if we move ny suset of point msses in the system to the enter of mss of this suset In other words, we n reple ny numer of point msses with single point mss, whose mss equls the sum of ll these msses nd whih is positioned t their M Given the oordinte system with the origin, we n speify position of ny geometri point y the vetor, onneting the origin with this point For the system of point msses,, loted t geometri points, position of point mss is speified y the vetor onneting the origin with point where the mss is loted It n e esily proven using the M definition given ove tht the position of the M of the system, M, is given y
5 , or, n importnt property of the M immeditely follows from the ove If we dd point to the system, the resultnt M is the M of the system of two points: the new point nd the point with mss pled t the M of the first n points, Prolem Prove tht the medins of n ritrry tringle re onurrent (ross t the sme point M) Prolem Prove tht the ltitudes of n ritrry tringle re onurrent (ross t the sme point H) Prolem Prove tht the isetors of n ritrry tringle re onurrent (ross t the sme point ) Prolem Pr ve ev the rem M
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