VIII - Geometric Vectors

Transcription

1 MTHEMTIS 0-05-RE Linea lgeba Matin Huad Fall 05 VIII - Geometic Vectos. Find all ectos in the following paallelepiped that ae equialent to the gien ectos. E F H G a) b) HE c) H 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. Let EF be a egula hexagon whee a and F b. a) Expess the othe sides,,, E and EF, in tems of a and b. b) Expess F, F and F in tems of a and b.. Poe that if then. 4. In the following paallelogam, M and N ae the midpoints of and espectiely. Poe that MN is a paallelogam. M N 5. Poe that if the midpoints of the adjacent sides in a ectangle ae joined, the esulting figue is a hombus (a paallelogam whose sides all hae equal length). 6. Let be a paallelogam. Veify that the diagonal and the line E, whee E is the midpoint of, intesect at a point that diides both of these segments in the atio of to.. Poe that the line segment joining the midpoints of the diagonals in a tapezoid is paallel to the base and is half the length of the diffeence between the lengths of the two bases.

2 Math Let be a paallelogam and let E diide the segment in a atio of to and let F diide the segment in two. In what atio does P, the point of intesection of F and E, diide the segments F and E? 9. Let be a tiangle and let diide the segment in a atio of to and let E diide the segment in a atio of to 4. In what atio does P, the point of intesection of E and, diide the segments E and? 0. etemine whethe the following statements ae tue of false. Explain a) Two equialent ectos hae the same initial point. b) 5 c) If then. d) and hae the same diection. e) and 5 hae the same diection. f) u and ku hae the same diection. g) If u k then u u. h) Thee exists two nonzeo ectos u and such that u u. j) i) u u Fall 05 Matin Huad

3 Math 05 nswes. a) EF HG b) GF c) G d) F G e) EG f) H E g) 0 h) FH i) EG j) E F H G k) G l) 0 m) E F H G n) G H o) E H. a) a b b E a EF b a b) F a b F a a b. Thus 4. We need to show that M N and N M. Fo Fo M N, M N N M, we hae M M since M is the midpoint of since is a paallelogam since N is the midpoint of M since M is the midpoint of Since N since M N and is a paallelogam N N N since N is the midpoint of Fall 05 Matin Huad

4 Math We need to show that PQRS is a paallelogam 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 ectangle and P,Q,R and S ae 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 ae of equal length, it suffices to show that PQ poed that PQRS is a paallelogam. PQ P Q R (by the Pythagoean theoem) PS since we hae P S PS (by the Pythagoean theoem) hence PQ PS 6. We need to show that if P E then and if P s then s. P Since is a paallelogam and E is the midpoint of, then E Let us expess P in tems of and in two diffeent ways. P E P s E E -s - s s s s s s s y the basis theoem, we hae the equations s, s. Soling, we obtain and s. Fall 05 Matin Huad 4

5 Math 05. We need to show that MN MN M N since M and N ae the midpoints of and 8. If P F and P se, then we need to find and s. Since is a paallelogam, we hae and. E P lso, by the definitions of E and F, we F hae s E and F. P F F Let us expess P in tems of and in two diffeent ways. P P se s E s y the basis theoem, we hae the equations s M 4 Soling, we obtain and s E in a atio of to 4. N s s s. Hence, P diides F in a atio of 4 to and diides Fall 05 Matin Huad 5

6 Math If P E and P s, then we need to find and s. y the definitions of and E, we hae 4 5 and E. E Let us expess P in tems of and in two diffeent ways. s P E E 4 s y the basis theoem, we hae the equations 4 s 0 Soling, we obtain and 9 s 9 diides in a atio of 0 to 9. P P 5 s s s s s 5 5. Hence, P diides E in a atio of to 8 and 0. a) F. They may hae any initial point as long as they hae the same diections and magnitude. b) T. 0 so 0 0. c) F. If, and hae opposite diection then. d) F. They hae opposite diection. 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 diection. Fall 05 Matin Huad 6