www.gshqxx.com:高中数学第1部分第二章章末小结课件北师大版必修

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第 二 章

章 末 小 结

核心要点归纳

阶段质量检测

1.平面向量有关的概念 (1)零向量和任何向量共线,与任一向量垂直,但 方向不定. (2)单位向量:长度为 1 个单位长度的向量叫做单 a 位向量.a 方向上的单位向量 a0=|a|. (3)平行(共线)向量:方向相同或相反,无传递性.

(4)相等向量:大小相等方向相同. (5)相反向量:大小相等方向相反. a· b (6)向量的射影:a 在 b 上的射影是|a|cos θ= (θ 为 |b| a,b 的夹角).

2.向量共线与平面向量基本定理

(1)两个非零向量平行(共线):a∥b?a=λb.
(2)平面向量基本定理: 若e1,e2是同一平面内的两个不共线向量,那么对于平 面内的任一向量a有且只有一对实数λ1,λ2使a=λ1e1+ λ2 e2 .

(3)三点 A,B,C 共线? AB , AC 共线: ??? ??? ??? ? 向量 PA, PB , PC 中三个终点 A,B,C 共线? ??? ??? ??? ? 存在实数 x,y,使 PA=x PB +y PC ,且 x+y=1.

??? ?

??? ?

3.向量的线性运算 (1)向量加法的三角形法则适用于任意两个非零向量 相加,并且可以推广到两个以上的非零向量相加;

(2)向量减法是加法的逆运算;
(3)向量平行(共线)与直线平行是有区别的,直线平行 不包括重合的情况.

4.向量的数量积 (1)平面向量的数量积: 几何表示 a· b=|a||b|cos θ(a≠0,b≠0,0° ≤θ≤180° ),0· a=0 坐标表示 a· b=x1x2+y1y2 a· b=b· a;

运算律

(λa)· b=a· (λb)
(a+b)· c=a· c+b· c

(2)平面向量数量积的重要性质:
(1)|a|= a· a= |a||a|cos θ= a· b 几何 (2)cos θ= ; |a||b| 关系 (3)|a· b|≤|a||b|; (4)a⊥b?a· b=0 |a|2;

2 (1)|a|= x2 + y 1 1;

坐标 表示

x1x2+y1y2 (2)cos θ= 2 2 2 2; x1+y1· x2+y2 (3)|x1x2+y1y2|≤
2 2 2 x2 + y · x + y 1 1 2 2;

(4)a· b?x1x2+y1y2=0


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