2022年《平面向量》知识点归纳总结 .pdf

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1、第一章平面向量2.1 向量的基本概念和基本运算16、向量：既有大小，又有方向的量数量：只有大小，没有方向的量有向线段的三要素：起点、方向、长度零向量：长度为0的向量单位向量：长度等于1个单位的向量平行向量（共线向量）：方向相同或相反的非零向量零向量与任一向量平行相等向量：长度相等且方向相同的向量17、向量加法运算：三角形法则的特点：首尾相连平行四边形法则的特点：共起点三角形不等式：ababab运算性质：交换律：abba；结合律：abcabc；00aaa坐标运算：设11,ax y，22,bxy，则1212,abxxyy18、向量减法运算：三角形法则的特点：共起点，连终点，方向指向被减向量坐标运算

2、：设11,ax y，22,bxy，则1212,abxxyy设、两点的坐标分别为11,x y，22,xy，则1212,xx yy19、向量数乘运算：实数与向量a的积是一个向量的运算叫做向量的数乘，记作aaa；当0时，a的方向与a的方向相同；当0时，a的方向与a的方向相反；当0时，0a运算律：aa；aaa；abab坐标运算：设,ax y，则,ax yxy20、向量共线定理：向量0a a与b共线，当且仅当有唯一一个实数，使ba设11,ax y，22,bxy，其中0b，则当且仅当12210 x yx y时，向量a、0b b共线2.2 平面向量的基本定理及坐标表示21、平面向量基本定理：如果1e、2e是

3、同一平面内 的两个 不共线 向量，那么对于这一平面内的任意向量a，有且只有一对实数1、2，使1 122aee（不共线的向量1e、2e作baCabCC为这一平面内所有向量的一组基底）22、分点坐标公式：设点是线段12上的一点，1、2的坐标分别是11,x y，22,xy，当12时，点的坐标是1212,11xxyy（当时，就为中点公式。）12.3 平面向量的数量积23、平面向量的数量积（两个向量的数量积等于它们对应坐标的乘积的和。）：cos0,0,0180a ba bab零向量与任一向量的数量积为0性质：设a和b都是非零向量，则0aba b 当a与b同向时，a ba b；当a与b反向时，a ba b

4、；22a aaa或aa aa ba b运算律：a bb a；aba bab；abca cb c坐标运算：设两个非零向量11,ax y，22,bxy，则1212a bx xy y若,ax y，则222axy，或22axy设11,ax y，22,bxy，则12120abx xy y设a、b都 是 非 零 向 量，11,ax y，22,bxy，是a与b的 夹 角，则121222221122cosx xy ya ba bxyxy知识链接：空间向量空间向量的许多知识可由平面向量的知识类比而得.下面对空间向量在立体几何中证明，求值的应用进行总结归纳.1、直线的方向向量和平面的法向量直线的方向向量：若 A、

5、B 是直线l上的任意两点，则AB为直线l的一个方向向量；与AB平行的任意非零向量也是直线l的方向向量.平面的法向量：若向量n所在直线垂直于平面，则称这个向量垂直于平面，记作n，如果n，那么向量n叫做平面的法向量.平面的法向量的求法（待定系数法）：建立适当的坐标系设平面的法向量为(,)nx y z文档编码:CQ4V5M10P5Y2 HQ3P4Z1Z7P5 ZA6P7C5T7A5文档编码:CQ4V5M10P5Y2 HQ3P4Z1Z7P5 ZA6P7C5T7A5文档编码:CQ4V5M10P5Y2 HQ3P4Z1Z7P5 ZA6P7C5T7A5文档编码:CQ4V5M10P5Y2 HQ3P4Z1Z7P5

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12、Z7P5 ZA6P7C5T7A5求出平面内两个不共线向量的坐标123123(,),(,)aa aabb b b根据法向量定义建立方程组00n an b.解方程组，取其中一组解，即得平面的法向量.（如图）1、用向量方法判定空间中的平行关系线线平行设直线12,l l的方向向量分别是a b、，则要证明1l2l，只需证明ab，即()akb kR.即：两直线平行或重合两直线的方向向量共线。线面平行（法一）设直线l的方向向量是a，平面的法向量是u，则要证明l，只需证明au，即0a u.即：直线与平面平行直线的方向向量与该平面的法向量垂直且直线在平面外（法二）要证明一条直线和一个平面平行，也可以在平面内找一

13、个向量与已知直线的方向向量是共线向量即可.面面平行若平面的法向量为u，平面的法向量为v，要证，只需证uv，即证uv.即：两平面平行或重合两平面的法向量共线。3、用向量方法判定空间的垂直关系线线垂直设直线12,l l的方向向量分别是a b、，则要证明12ll，只需证明ab，即0a b.即：两直线垂直两直线的方向向量垂直。线面垂直（法一）设直线l的方向向量是a，平面的法向量是u，则要证明l，只需证明au，即au.（法二）设直线l的方向向量是a，平面内的两个相交向量分别为mn、，若文档编码:CQ4V5M10P5Y2 HQ3P4Z1Z7P5 ZA6P7C5T7A5文档编码:CQ4V5M10P5Y2 H

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19、编码:CQ4V5M10P5Y2 HQ3P4Z1Z7P5 ZA6P7C5T7A5文档编码:CQ4V5M10P5Y2 HQ3P4Z1Z7P5 ZA6P7C5T7A5文档编码:CQ4V5M10P5Y2 HQ3P4Z1Z7P5 ZA6P7C5T7A5文档编码:CQ4V5M10P5Y2 HQ3P4Z1Z7P5 ZA6P7C5T7A5文档编码:CQ4V5M10P5Y2 HQ3P4Z1Z7P5 ZA6P7C5T7A5文档编码:CQ4V5M10P5Y2 HQ3P4Z1Z7P5 ZA6P7C5T7A5文档编码:CQ4V5M10P5Y2 HQ3P4Z1Z7P5 ZA6P7C5T7A5文档编码:CQ4V5M10P5

20、Y2 HQ3P4Z1Z7P5 ZA6P7C5T7A5文档编码:CQ4V5M10P5Y2 HQ3P4Z1Z7P5 ZA6P7C5T7A5文档编码:CQ4V5M10P5Y2 HQ3P4Z1Z7P5 ZA6P7C5T7A50,.0a mla n则即：直线与平面垂直直线的方向向量与平面的法向量共线直线的方向向量与平面内两条不共线直线的方向向量都垂直。面面垂直若平面的法向量为u，平面的法向量为v，要证，只需证uv，即证0u v.即：两平面垂直两平面的法向量垂直。4、利用向量求空间角求异面直线所成的角已知,a b为两异面直线，A，C 与 B，D 分别是,a b上的任意两点，,a b所成的角为，则cos.A

21、C BDAC BD求直线和平面所成的角 定义：平面的一条斜线和它在平面上的射影所成的锐角叫做这条斜线和这个平面所成的角求法：设直线l的方向向量为a，平面的法向量为u，直线与平面所成的角为，a与u的夹角为，则为的余角或的补角的余角.即有：coss.ina ua u求二面角 定义：平面内的一条直线把平面分为两个部分，其中的每一部分叫做半平面；从一条直线出发的两个半平面所组成的图形叫做二面角，这条直线叫做二面角的棱，每个半平面叫做二面角的面二面角的平面角是指在二面角l的棱上任取一点O，分别在两个半平面内作射线lBOlAO,，则AOB为二面角l的平面角.如图：求法：设二面角l的两个半平面的法向量分别为

22、m n、，再设m n、的夹角为，二面角l的平面角为，则二面角为m n、的夹角或其补角.O A B O A B l 文档编码:CQ4V5M10P5Y2 HQ3P4Z1Z7P5 ZA6P7C5T7A5文档编码:CQ4V5M10P5Y2 HQ3P4Z1Z7P5 ZA6P7C5T7A5文档编码:CQ4V5M10P5Y2 HQ3P4Z1Z7P5 ZA6P7C5T7A5文档编码:CQ4V5M10P5Y2 HQ3P4Z1Z7P5 ZA6P7C5T7A5文档编码:CQ4V5M10P5Y2 HQ3P4Z1Z7P5 ZA6P7C5T7A5文档编码:CQ4V5M10P5Y2 HQ3P4Z1Z7P5 ZA6P7C5T

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26、5T7A5文档编码:CQ4V5M10P5Y2 HQ3P4Z1Z7P5 ZA6P7C5T7A5文档编码:CQ4V5M10P5Y2 HQ3P4Z1Z7P5 ZA6P7C5T7A5文档编码:CQ4V5M10P5Y2 HQ3P4Z1Z7P5 ZA6P7C5T7A5文档编码:CQ4V5M10P5Y2 HQ3P4Z1Z7P5 ZA6P7C5T7A5文档编码:CQ4V5M10P5Y2 HQ3P4Z1Z7P5 ZA6P7C5T7A5文档编码:CQ4V5M10P5Y2 HQ3P4Z1Z7P5 ZA6P7C5T7A5文档编码:CQ4V5M10P5Y2 HQ3P4Z1Z7P5 ZA6P7C5T7A5文档编码:CQ4

27、V5M10P5Y2 HQ3P4Z1Z7P5 ZA6P7C5T7A5文档编码:CQ4V5M10P5Y2 HQ3P4Z1Z7P5 ZA6P7C5T7A5文档编码:CQ4V5M10P5Y2 HQ3P4Z1Z7P5 ZA6P7C5T7A5文档编码:CQ4V5M10P5Y2 HQ3P4Z1Z7P5 ZA6P7C5T7A5文档编码:CQ4V5M10P5Y2 HQ3P4Z1Z7P5 ZA6P7C5T7A5文档编码:CQ4V5M10P5Y2 HQ3P4Z1Z7P5 ZA6P7C5T7A5文档编码:CQ4V5M10P5Y2 HQ3P4Z1Z7P5 ZA6P7C5T7A5文档编码:CQ4V5M10P5Y2 HQ3

28、P4Z1Z7P5 ZA6P7C5T7A5文档编码:CQ4V5M10P5Y2 HQ3P4Z1Z7P5 ZA6P7C5T7A5文档编码:CQ4V5M10P5Y2 HQ3P4Z1Z7P5 ZA6P7C5T7A5文档编码:CQ4V5M10P5Y2 HQ3P4Z1Z7P5 ZA6P7C5T7A5文档编码:CQ4V5M10P5Y2 HQ3P4Z1Z7P5 ZA6P7C5T7A5文档编码:CQ4V5M10P5Y2 HQ3P4Z1Z7P5 ZA6P7C5T7A5根据具体图形确定是锐角或是钝角：如果是锐角，则coscosm nm n，即arccosm nm n；如果是钝角，则coscosm nm n，即arcc

29、osm nm n.5、利用法向量求空间距离点 Q到直线l距离若 Q为直线l外的一点,P在直线l上，a为直线l的方向向量，b=PQ，则点 Q到直线l距离为221(|)()|haba ba点 A 到平面的距离若点 P 为平面外一点，点M 为平面内任一点，平面的法向量为n，则 P 到平面的距离就等于MP在法向量n方向上的投影的绝对值.即cos,dMPn MPnM PMPn MPn MPn直线a与平面之间的距离当一条直线和一个平面平行时，直线上的各点到平面的距离相等。由此可知，直线到平面的距离可转化为求直线上任一点到平面的距离，即转化为点面距离。即.n MPdn两平行平面,之间的距离文档编码:CQ4V

30、5M10P5Y2 HQ3P4Z1Z7P5 ZA6P7C5T7A5文档编码:CQ4V5M10P5Y2 HQ3P4Z1Z7P5 ZA6P7C5T7A5文档编码:CQ4V5M10P5Y2 HQ3P4Z1Z7P5 ZA6P7C5T7A5文档编码:CQ4V5M10P5Y2 HQ3P4Z1Z7P5 ZA6P7C5T7A5文档编码:CQ4V5M10P5Y2 HQ3P4Z1Z7P5 ZA6P7C5T7A5文档编码:CQ4V5M10P5Y2 HQ3P4Z1Z7P5 ZA6P7C5T7A5文档编码:CQ4V5M10P5Y2 HQ3P4Z1Z7P5 ZA6P7C5T7A5文档编码:CQ4V5M10P5Y2 HQ3P

31、4Z1Z7P5 ZA6P7C5T7A5文档编码:CQ4V5M10P5Y2 HQ3P4Z1Z7P5 ZA6P7C5T7A5文档编码:CQ4V5M10P5Y2 HQ3P4Z1Z7P5 ZA6P7C5T7A5文档编码:CQ4V5M10P5Y2 HQ3P4Z1Z7P5 ZA6P7C5T7A5文档编码:CQ4V5M10P5Y2 HQ3P4Z1Z7P5 ZA6P7C5T7A5文档编码:CQ4V5M10P5Y2 HQ3P4Z1Z7P5 ZA6P7C5T7A5文档编码:CQ4V5M10P5Y2 HQ3P4Z1Z7P5 ZA6P7C5T7A5文档编码:CQ4V5M10P5Y2 HQ3P4Z1Z7P5 ZA6P7

32、C5T7A5文档编码:CQ4V5M10P5Y2 HQ3P4Z1Z7P5 ZA6P7C5T7A5文档编码:CQ4V5M10P5Y2 HQ3P4Z1Z7P5 ZA6P7C5T7A5文档编码:CQ4V5M10P5Y2 HQ3P4Z1Z7P5 ZA6P7C5T7A5文档编码:CQ4V5M10P5Y2 HQ3P4Z1Z7P5 ZA6P7C5T7A5文档编码:CQ4V5M10P5Y2 HQ3P4Z1Z7P5 ZA6P7C5T7A5文档编码:CQ4V5M10P5Y2 HQ3P4Z1Z7P5 ZA6P7C5T7A5文档编码:CQ4V5M10P5Y2 HQ3P4Z1Z7P5 ZA6P7C5T7A5文档编码:CQ

33、4V5M10P5Y2 HQ3P4Z1Z7P5 ZA6P7C5T7A5文档编码:CQ4V5M10P5Y2 HQ3P4Z1Z7P5 ZA6P7C5T7A5文档编码:CQ4V5M10P5Y2 HQ3P4Z1Z7P5 ZA6P7C5T7A5文档编码:CQ4V5M10P5Y2 HQ3P4Z1Z7P5 ZA6P7C5T7A5文档编码:CQ4V5M10P5Y2 HQ3P4Z1Z7P5 ZA6P7C5T7A5文档编码:CQ4V5M10P5Y2 HQ3P4Z1Z7P5 ZA6P7C5T7A5文档编码:CQ4V5M10P5Y2 HQ3P4Z1Z7P5 ZA6P7C5T7A5文档编码:CQ4V5M10P5Y2 HQ

34、3P4Z1Z7P5 ZA6P7C5T7A5文档编码:CQ4V5M10P5Y2 HQ3P4Z1Z7P5 ZA6P7C5T7A5文档编码:CQ4V5M10P5Y2 HQ3P4Z1Z7P5 ZA6P7C5T7A5文档编码:CQ4V5M10P5Y2 HQ3P4Z1Z7P5 ZA6P7C5T7A5文档编码:CQ4V5M10P5Y2 HQ3P4Z1Z7P5 ZA6P7C5T7A5文档编码:CQ4V5M10P5Y2 HQ3P4Z1Z7P5 ZA6P7C5T7A5文档编码:CQ4V5M10P5Y2 HQ3P4Z1Z7P5 ZA6P7C5T7A5文档编码:CQ4V5M10P5Y2 HQ3P4Z1Z7P5 ZA6

35、P7C5T7A5文档编码:CQ4V5M10P5Y2 HQ3P4Z1Z7P5 ZA6P7C5T7A5文档编码:CQ4V5M10P5Y2 HQ3P4Z1Z7P5 ZA6P7C5T7A5文档编码:CQ4V5M10P5Y2 HQ3P4Z1Z7P5 ZA6P7C5T7A5文档编码:CQ4V5M10P5Y2 HQ3P4Z1Z7P5 ZA6P7C5T7A5文档编码:CQ4V5M10P5Y2 HQ3P4Z1Z7P5 ZA6P7C5T7A5文档编码:CQ4V5M10P5Y2 HQ3P4Z1Z7P5 ZA6P7C5T7A5文档编码:CQ4V5M10P5Y2 HQ3P4Z1Z7P5 ZA6P7C5T7A5文档编码:

36、CQ4V5M10P5Y2 HQ3P4Z1Z7P5 ZA6P7C5T7A5文档编码:CQ4V5M10P5Y2 HQ3P4Z1Z7P5 ZA6P7C5T7A5文档编码:CQ4V5M10P5Y2 HQ3P4Z1Z7P5 ZA6P7C5T7A5文档编码:CQ4V5M10P5Y2 HQ3P4Z1Z7P5 ZA6P7C5T7A5利用两平行平面间的距离处处相等，可将两平行平面间的距离转化为求点面距离。即.n MPdn异面直线间的距离设向量n与两异面直线,a b都垂直，,Ma Pb则两异面直线,a b间的距离d就是MP在向量n方向上投影的绝对值。即.n MPdn6、三垂线定理及其逆定理三垂线定理：在平面内的一

37、条直线，如果它和这个平面的一条斜线的射影垂直，那么它也和这条斜线垂直推理模式：,POOPAAaPAaaOAaPOA概括为：垂直于射影就垂直于斜线.三垂线定理的逆定理：在平面内的一条直线，如果和这个平面的一条斜线垂直，那么它也和这条斜线的射影垂直推理模式：,POOPAAaAOaaAP概括为：垂直于斜线就垂直于射影.7、三余弦定理设 AC是平面内的任一条直线，AD是的一条斜线AB在内的射影，且 BD AD，垂足为 D.设 AB与(AD)所成的角为1，AD 与 AC所成的角为2，AB 与 AC所成的角为 则12coscoscos.21ABDC文档编码:CQ4V5M10P5Y2 HQ3P4Z1Z7P5

38、 ZA6P7C5T7A5文档编码:CQ4V5M10P5Y2 HQ3P4Z1Z7P5 ZA6P7C5T7A5文档编码:CQ4V5M10P5Y2 HQ3P4Z1Z7P5 ZA6P7C5T7A5文档编码:CQ4V5M10P5Y2 HQ3P4Z1Z7P5 ZA6P7C5T7A5文档编码:CQ4V5M10P5Y2 HQ3P4Z1Z7P5 ZA6P7C5T7A5文档编码:CQ4V5M10P5Y2 HQ3P4Z1Z7P5 ZA6P7C5T7A5文档编码:CQ4V5M10P5Y2 HQ3P4Z1Z7P5 ZA6P7C5T7A5文档编码:CQ4V5M10P5Y2 HQ3P4Z1Z7P5 ZA6P7C5T7A5文

39、档编码:CQ4V5M10P5Y2 HQ3P4Z1Z7P5 ZA6P7C5T7A5文档编码:CQ4V5M10P5Y2 HQ3P4Z1Z7P5 ZA6P7C5T7A5文档编码:CQ4V5M10P5Y2 HQ3P4Z1Z7P5 ZA6P7C5T7A5文档编码:CQ4V5M10P5Y2 HQ3P4Z1Z7P5 ZA6P7C5T7A5文档编码:CQ4V5M10P5Y2 HQ3P4Z1Z7P5 ZA6P7C5T7A5文档编码:CQ4V5M10P5Y2 HQ3P4Z1Z7P5 ZA6P7C5T7A5文档编码:CQ4V5M10P5Y2 HQ3P4Z1Z7P5 ZA6P7C5T7A5文档编码:CQ4V5M10P

40、5Y2 HQ3P4Z1Z7P5 ZA6P7C5T7A5文档编码:CQ4V5M10P5Y2 HQ3P4Z1Z7P5 ZA6P7C5T7A5文档编码:CQ4V5M10P5Y2 HQ3P4Z1Z7P5 ZA6P7C5T7A5文档编码:CQ4V5M10P5Y2 HQ3P4Z1Z7P5 ZA6P7C5T7A5文档编码:CQ4V5M10P5Y2 HQ3P4Z1Z7P5 ZA6P7C5T7A5文档编码:CQ4V5M10P5Y2 HQ3P4Z1Z7P5 ZA6P7C5T7A5文档编码:CQ4V5M10P5Y2 HQ3P4Z1Z7P5 ZA6P7C5T7A5文档编码:CQ4V5M10P5Y2 HQ3P4Z1Z7

41、P5 ZA6P7C5T7A5文档编码:CQ4V5M10P5Y2 HQ3P4Z1Z7P5 ZA6P7C5T7A5文档编码:CQ4V5M10P5Y2 HQ3P4Z1Z7P5 ZA6P7C5T7A5文档编码:CQ4V5M10P5Y2 HQ3P4Z1Z7P5 ZA6P7C5T7A5文档编码:CQ4V5M10P5Y2 HQ3P4Z1Z7P5 ZA6P7C5T7A5文档编码:CQ4V5M10P5Y2 HQ3P4Z1Z7P5 ZA6P7C5T7A5文档编码:CQ4V5M10P5Y2 HQ3P4Z1Z7P5 ZA6P7C5T7A5文档编码:CQ4V5M10P5Y2 HQ3P4Z1Z7P5 ZA6P7C5T7A

42、5文档编码:CQ4V5M10P5Y2 HQ3P4Z1Z7P5 ZA6P7C5T7A5文档编码:CQ4V5M10P5Y2 HQ3P4Z1Z7P5 ZA6P7C5T7A5文档编码:CQ4V5M10P5Y2 HQ3P4Z1Z7P5 ZA6P7C5T7A5文档编码:CQ4V5M10P5Y2 HQ3P4Z1Z7P5 ZA6P7C5T7A5文档编码:CQ4V5M10P5Y2 HQ3P4Z1Z7P5 ZA6P7C5T7A5文档编码:CQ4V5M10P5Y2 HQ3P4Z1Z7P5 ZA6P7C5T7A5文档编码:CQ4V5M10P5Y2 HQ3P4Z1Z7P5 ZA6P7C5T7A5文档编码:CQ4V5M1

43、0P5Y2 HQ3P4Z1Z7P5 ZA6P7C5T7A5文档编码:CQ4V5M10P5Y2 HQ3P4Z1Z7P5 ZA6P7C5T7A5文档编码:CQ4V5M10P5Y2 HQ3P4Z1Z7P5 ZA6P7C5T7A5文档编码:CQ4V5M10P5Y2 HQ3P4Z1Z7P5 ZA6P7C5T7A5文档编码:CQ4V5M10P5Y2 HQ3P4Z1Z7P5 ZA6P7C5T7A5文档编码:CQ4V5M10P5Y2 HQ3P4Z1Z7P5 ZA6P7C5T7A5文档编码:CQ4V5M10P5Y2 HQ3P4Z1Z7P5 ZA6P7C5T7A5文档编码:CQ4V5M10P5Y2 HQ3P4Z1

44、Z7P5 ZA6P7C5T7A5文档编码:CQ4V5M10P5Y2 HQ3P4Z1Z7P5 ZA6P7C5T7A5文档编码:CQ4V5M10P5Y2 HQ3P4Z1Z7P5 ZA6P7C5T7A5文档编码:CQ4V5M10P5Y2 HQ3P4Z1Z7P5 ZA6P7C5T7A58、面积射影定理已知平面内一个多边形的面积为S S原，它在平面内的射影图形的面积为SS射，平面与平面所成的二面角的大小为锐二面角，则cos=.SSSS射原9、一个结论长度为l的线段在三条两两互相垂直的直线上的射影长分别为123lll、，夹角分别为123、,则有2222123llll222123coscoscos12221

45、23sinsinsin2.（立体几何中长方体对角线长的公式是其特例）.文档编码:CQ4V5M10P5Y2 HQ3P4Z1Z7P5 ZA6P7C5T7A5文档编码:CQ4V5M10P5Y2 HQ3P4Z1Z7P5 ZA6P7C5T7A5文档编码:CQ4V5M10P5Y2 HQ3P4Z1Z7P5 ZA6P7C5T7A5文档编码:CQ4V5M10P5Y2 HQ3P4Z1Z7P5 ZA6P7C5T7A5文档编码:CQ4V5M10P5Y2 HQ3P4Z1Z7P5 ZA6P7C5T7A5文档编码:CQ4V5M10P5Y2 HQ3P4Z1Z7P5 ZA6P7C5T7A5文档编码:CQ4V5M10P5Y2 H

46、Q3P4Z1Z7P5 ZA6P7C5T7A5文档编码:CQ4V5M10P5Y2 HQ3P4Z1Z7P5 ZA6P7C5T7A5文档编码:CQ4V5M10P5Y2 HQ3P4Z1Z7P5 ZA6P7C5T7A5文档编码:CQ4V5M10P5Y2 HQ3P4Z1Z7P5 ZA6P7C5T7A5文档编码:CQ4V5M10P5Y2 HQ3P4Z1Z7P5 ZA6P7C5T7A5文档编码:CQ4V5M10P5Y2 HQ3P4Z1Z7P5 ZA6P7C5T7A5文档编码:CQ4V5M10P5Y2 HQ3P4Z1Z7P5 ZA6P7C5T7A5文档编码:CQ4V5M10P5Y2 HQ3P4Z1Z7P5 ZA

47、6P7C5T7A5文档编码:CQ4V5M10P5Y2 HQ3P4Z1Z7P5 ZA6P7C5T7A5文档编码:CQ4V5M10P5Y2 HQ3P4Z1Z7P5 ZA6P7C5T7A5文档编码:CQ4V5M10P5Y2 HQ3P4Z1Z7P5 ZA6P7C5T7A5文档编码:CQ4V5M10P5Y2 HQ3P4Z1Z7P5 ZA6P7C5T7A5文档编码:CQ4V5M10P5Y2 HQ3P4Z1Z7P5 ZA6P7C5T7A5文档编码:CQ4V5M10P5Y2 HQ3P4Z1Z7P5 ZA6P7C5T7A5文档编码:CQ4V5M10P5Y2 HQ3P4Z1Z7P5 ZA6P7C5T7A5文档编码

48、:CQ4V5M10P5Y2 HQ3P4Z1Z7P5 ZA6P7C5T7A5文档编码:CQ4V5M10P5Y2 HQ3P4Z1Z7P5 ZA6P7C5T7A5文档编码:CQ4V5M10P5Y2 HQ3P4Z1Z7P5 ZA6P7C5T7A5文档编码:CQ4V5M10P5Y2 HQ3P4Z1Z7P5 ZA6P7C5T7A5文档编码:CQ4V5M10P5Y2 HQ3P4Z1Z7P5 ZA6P7C5T7A5文档编码:CQ4V5M10P5Y2 HQ3P4Z1Z7P5 ZA6P7C5T7A5文档编码:CQ4V5M10P5Y2 HQ3P4Z1Z7P5 ZA6P7C5T7A5文档编码:CQ4V5M10P5Y2

49、 HQ3P4Z1Z7P5 ZA6P7C5T7A5文档编码:CQ4V5M10P5Y2 HQ3P4Z1Z7P5 ZA6P7C5T7A5文档编码:CQ4V5M10P5Y2 HQ3P4Z1Z7P5 ZA6P7C5T7A5文档编码:CQ4V5M10P5Y2 HQ3P4Z1Z7P5 ZA6P7C5T7A5文档编码:CQ4V5M10P5Y2 HQ3P4Z1Z7P5 ZA6P7C5T7A5文档编码:CQ4V5M10P5Y2 HQ3P4Z1Z7P5 ZA6P7C5T7A5文档编码:CQ4V5M10P5Y2 HQ3P4Z1Z7P5 ZA6P7C5T7A5文档编码:CQ4V5M10P5Y2 HQ3P4Z1Z7P5

50、ZA6P7C5T7A5文档编码:CQ4V5M10P5Y2 HQ3P4Z1Z7P5 ZA6P7C5T7A5文档编码:CQ4V5M10P5Y2 HQ3P4Z1Z7P5 ZA6P7C5T7A5文档编码:CQ4V5M10P5Y2 HQ3P4Z1Z7P5 ZA6P7C5T7A5文档编码:CQ4V5M10P5Y2 HQ3P4Z1Z7P5 ZA6P7C5T7A5文档编码:CQ4V5M10P5Y2 HQ3P4Z1Z7P5 ZA6P7C5T7A5文档编码:CQ4V5M10P5Y2 HQ3P4Z1Z7P5 ZA6P7C5T7A5文档编码:CQ4V5M10P5Y2 HQ3P4Z1Z7P5 ZA6P7C5T7A5文档