Question

(本小題滿分14分)
已知向量a =" (" sinx , 0 ), b =" (cosx," 1), 其中 0 < x <, 求|a –b |的取值范圍

Answer 1

|a - b |2 Î [, )

解1:  |a –b |2 =" |" (cosx–sinx, ) |2             2分
= (cosx–sinx)2 +                           3 分
=" sin2(x" – ) +.                                 3分
? 0 < x < , ∴–< x - < ,                          2分
∴ 0 £ sin2(C– ) < ,                                       2分
得 |a –b | Î [, ).                                 2分
解2:  |a –b |2 = | a |2 – a·b + | b |2              2分 
= sin2–sinxcosx + (cos2x +1)                      2分
=sin2–sinxcosx + cos2x +
= (cosx – sinx)2 +                               2 分
=" sin2(x" – ) +.                                     2分
? 0 < x < , ∴–< x - < ,                        2分
∴ 0 £ sin2(C– ) < ,                                      2分
得  |a - b |2 Î [, ).                            2分