MTHEMTIS 0-NY-05 Vectors and Matrices Martin Huard Fall 07 VIII - Geometric Vectors. Find all ectors in the following parallelepiped that are equialent to the gien ectors. E F H G a) b) c) d) E e) f) F g) HE h) F G i) FG FE j) E k) E l) m) H H E n) GH E E F o) G E G HE H. Let EF be a regular hexagon where a and F b. a) Express the other sides,,, E and EF, in terms of a and b b) Express F, F and F in terms of a and b... Let EFGH be a regular octagon where a and b. a) Express all other sides in terms of a and b. b) Express,, E and H in terms of a and b. 4. onsider the ectors u, and w such that u 4, N5 E, S45 W w 6, N60 W Find the length and direction of the following ectors. a) u b) u w c) w u d) u e) w 5
Math NY 5. n airplane has a maximum air speed of 500 km/h. If the plane is flying at is maximum speed with a heading of 40 degrees west of north and the wind is blowing from north to south at 40 km/h, find the ground speed of the aircraft and the direction. 6. jet if flying through a wind that is blowing with a sped of 40 km/h in the direction N0 E. The jet has a speed of 60 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. woman launches a boat from on shore of a straight rier and wants to land at the point directly on the opposite shore. If the speed of the boat (in still water) Is 0 km/h and the rier is flowing east at the rate of 5 km/h, in what direction should she head the boat in order to arrie at the desired landing point? 8. boat heads in the direction N60 E. The speed of the boat in still water is 4 km/h. The water is flowing directly south. It is obsered that the true direction of the boat is directly east. Find the speed of the water and the true speed of the boat. 9. Proe that if then. 0. In the following parallelogram, M and N are the midpoints of and respectiely. Proe that MN is a parallelogram. M N. Let be any quadrilateral such that O O O O where O is any point. Proe that must be a parallelogram.. Proe that if the midpoints of the adjacent sides in a rectangle are joined, the resulting figure is a rhombus.. Let be a parallelogram. Verify that the diagonal and the line E, where E is the midpoint of, intersect at a point that diides both of these segments in the ratio of to. 4. Let be any quadrilateral on a plane. Proe that if P, Q, R and S are the midpoints of segments,, and respectiely, then PQRS is a parallelogram. 5. Proe that the line segment joining the midpoints of the diagonals in a trapezoid is parallel to the base and is half the length of the difference between the lengths of the two bases. Fall 07 Martin Huard
Math NY 6. Let be a parallelogram and let E diide the segment in a ratio of to and let F diide the segment in two. In what ratio does P, the point of intersection of F and E, diide the segments F and E? 7. etermine whether the following statements are true of false. Explain a) Two equialent ectors hae the same initial point. b) 5 c) If then. d) and hae the same direction. e) and f) If u k then u u. 5 hae the same direction. g) There exists two nonzero ectors u h) u u and such that u u. i) j) u and ku hae the same direction. Fall 07 Martin Huard
Math NY nswers. a) EF HG b) GF c) G d) F G e) EG f) H E g) 0 h) i) EG FH j) E F H G k) G l) 0 m) E F H G n) G H o) b E. a) a b b) F a b. a) b a E b a GH a b H a b b) a b F a a EF b a a b EF a b FG b E H E b a H b a 4. a) u 5, N W, arcsin 5 5. 6. b) c) 5 sin 5 5 cos 5 u w = cos, N W, =arcsin 5 sin 5 5 cos 5 w u = cos, S W, =arcsin 5 u = 5, N E, =arcsin 5 d) 5 e) w = 0 6 cos, S E, =60 arcsin 5 5 6 5 sin 5 06cos 5 sin 9 9 6900cos 9 0 69 00cos km/h, N W, arcsin 40 6sin 9 77488cos 0 77 488cos km/h, N E, arcsin 0 7. N0 W 8. km/h and km/h 9. Thus Since Fall 07 Martin Huard 4
Math NY 0. We need to show that M N and For N M M N, since is a parallelogram M M N N. M M N N since M and N are the midpoints of and M N For N M M N, we hae M M M since M is the midpoint of N since M N and is a parallelogram N since N is the midpoint of N N. We need to show that and. For we hae O O O O O O O O O O O O nd for, O O O O O O O O O O O O O O O O Fall 07 Martin Huard 5
Math NY. We need to show that PQRS is a parallelogram whose sides hae equal length. P PQ P S SR R Q P Q SR P Q P Q SR P Q SR Q S Since is a rectangle and P,Q,R and S are midpoints, then P P R R S S Q Q QR QP PS SR PQ PS SR SR PS SR PS To show that all sides are of equal length, it suffices to show that PQ proed that PQRS is a parallelogram. PQ P Q R (by the Pythagorean theorem) PS since we hae P S PS (by the Pythagorean theorem) hence PQ PS. We need to show that if P r E then r and if P s then s. r P Since is a parallelogram and E is the midpoint of, then E Let us express P in terms of and in two different ways. P r E P s r E r r r E -s -r s s s s s s s y the basis theorem, we hae the equations r s, r s. Soling, we obtain r and s. Fall 07 Martin Huard 6
Math NY 4. We need to show that PQ SR and QR PS. P Q R S For PQ SR PQ P S SR R Q P S SR R Q P S SR R Q P Q S R SR PQ SR SR Thus PQ SR and PQ SR. For QR PS, we hae QR QP PS SR PQ PS SR SR PS SR PS 5. We need to show that MN Since P and Q are the midpoints of and Since S and R are the midpoints of and y associatiity M N We hae MN M N and also MN M N M N since M and N are the midpoints of and M N dding, we obtain MN M M N N Thus MN Fall 07 Martin Huard 7
Math NY 6. If P r F and P se, then we need to find r and s. Since is a parallelogram, we hae and. E P lso, by the definitions of E and F, we F r hae s E and F. P r F r F r r r r Let us express P in terms of and in two different ways. P P y the basis theorem, we hae the equations r s 4 Soling, we obtain r and 7 s 7 E in a ratio of to 4. r s se s E s s s. Hence, P diides F in a ratio of 4 to and diides 7. a) F. They may hae any initial point as long as the hae the same directions and magnitude. b) T. 0 so 0 0. c) F. If, and hae opposite direction then. d) F. They hae opposite direction. e) T.. f) F. If k then u and we hae u 0 0 u g) T. If u then u and u h) T. u u u u i) T. j) F. If k then u and ku u hae opposite direction. Fall 07 Martin Huard 8