文档介绍：CHAPTER 1 . Given the vectors M =? 10 a x + 4 a y ? 8 a z and N = 8 a x + 7 a y ? 2 a z, ?nd: a) a unit vector in the direction of ? M + 2 N . ? M + 2 N = 10 a x ? 4 a y + 8 a z + 16 a x + 14 a y ? 4 a z = ( 26 , 10 , 4 ) Thus a = ( 26 , 10 , 4 ) | ( 26 , 10 , 4 ) | = ( 0 . 92 , 0 . 36 , 0 . 14 ) b) the magnitude of 5 a x + N ? 3 M : ( 5 , 0 , 0 ) + ( 8 , 7 , ? 2 ) ?( ? 30 , 12 , ? 24 ) = ( 43 , ? 5 , 22 ), and | ( 43 , ? 5 , 22 ) |= 48 . 6 . c) | M || 2 N | ( M + N ) : | ( ? 10 , 4 , ? 8 ) || ( 16 , 14 , ? 4 ) | ( ? 2 , 11 , ? 10 ) = ( 13 . 4 )( 21 . 6 )( ? 2 , 11 , ? 10 ) = ( ? 580 . 5 , 3193 , ? 2902 ) . Given three points, A( 4 , 3 , 2 ) , B( ? 2 , 0 , 5 ), and C( 7 , ? 2 , 1 ) : a) Specify the vector Aextending from the origin to the point A . A = ( 4 , 3 , 2 ) = 4 a x + 3 a y + 2 a z b) Give a unit vector extending from the origin to the midpoint of line AB . The vector from the origin to the midpoint is given by M = ( 1 / 2 )( A + B ) = ( 1 / 2 )( 4 ? 2 , 3 + 0 , 2 + 5 ) = ( 1 , 1 . 5 , 3 . 5 ) The unit vector will be m = ( 1 , 1 . 5 , 3 . 5 ) | ( 1 , 1 . 5 , 3 . 5 ) | = ( 0 . 25 , 0 . 38 , 0 . 89 ) c) Calculate the length of the perimeter of triangle ABC : Begin with AB = ( ? 6 , ? 3 , 3 ) , BC = ( 9 , ? 2 , ? 4 ) , CA = ( 3 , ? 5 , ? 1 ) . Then | AB |+| BC |+| CA |= 7 . 35 + 10 . 05 + 5 . 91 = 23 . 32 . The vector from the origin to the point Ais given as ( 6 , ? 2 , ? 4 ), and the unit vector directed from the origin toward point B is ( 2 , ? 2 , 1 )/3. If points A and Bare ten units apart, ?nd the coordinates of point B . With A = ( 6 , ? 2 , ? 4 ) and B = 1 3 B( 2 , ? 2 , 1 ), we use the fact that | B ? A |= 10, or | ( 6 ? 2 3 B) a x ?( 2 ? 2 3 B) a y ?( 4 + 1 3 B) a z |= 10 Expanding, obtain 36 ? 8 B + 4 9 B 2 + 4 ? 8 3 B + 4 9 B 2 + 16 + 8 3 B + 1 9 B 2 = 100 or B 2 ? 8 B ? 44 =0. Thus B = 8 ± √ 64 ? 176 2 = 11 .75 (taking positive option) and so B = 2 3 ( 11 . 75 ) a x ? 2 3 ( 11 . 75 ) a y + 1 3 ( 11 . 75 ) a z = 7 . 83 a x ? 7 . 83 a y + 3