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bifenzhibowang:2019-2020学年高中数学 第二章《平面向量数量积的坐标表示》教案 北师大版必修4.doc

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2019-2020 学年高中数学 第二章《平面向量数量积的坐标表示》教 案 北师大版必修 4
一.教学目标: 1.知识与技能 (1)掌握数量积的坐标表达式,会进行平面向量数量积的运算. (2)能运用数量积表示两个向量的夹角,会用数量积判断两个平面向量的垂直关系. (3)揭示知识背景,创设问题情景,强化学生的参与意识. 2.过程与方法 通过本节课的学习,让学生体会应用向量知识处理解析几 何问题是一种有效手段,通过应 用帮助学生掌握几个公式的等价形式,然后和同学一起总结方法,最后巩固强化. 3.情感态度价值观 通过 本节的学习,使同学们对用坐标来研究向量的数量积有了一个崭新的认识;提高学生 迁移知识的能力. 二.教学重、难点 重点: 平面向量数量积的坐标表示以及推得的长度、角度、垂直关系的坐标表示. 难点: 用坐标法处理长度、角度、垂直问题. 三.学法与教学用具 学法:(1)自主性学习法+探究式学习法 (2)反馈练习法: 以练习来检验知识的应用情况,找出未掌握的内容及其存在的差 距. 教学用具:电脑、投影机. 四.教学设想 【创 设情境】 [展示投影]引入: 请同学们回忆一下实数与向量的乘积的坐标表 示以及两向量共线的坐标表示: 【探究新知】 平面两向量数量积的坐标又如何表示呢? 1. 推导坐标公式:设 a = (x1, y1),b = (x2, y2),x 轴上单位向量 i,y 轴上单位向量 j, 则:i?i = 1,j?j = 1,i? j = j?i = 0. ∵a = x1i + y1j, b = x2i + y2j ∴a?b = (x1i + y1j )(x2i + y2j) = x1x2i2 + x1y1i?j + x2y1i?j + y1y2j2 = x1x2 + y1y2 从而获得公式: a?b = x1x2 + y1y2 2.长度、角度、垂直的坐标表示 ①a = (x, y) |a|2 = x2 + y2 |a| =
? ??

x2 ? y2

( x1 ? x 2 ) 2 ? ( y1 ? y 2 ) 2 ②若 A = (x1, y1),B = (x2, y2),则 AB =

③cos ④∵a

a?b ? =| a |?|b|
b

x1 x 2 ? y1 y 2 x1 ? y1
2 2

x2 ? y2

2

2

a?b = 0 即 x1x2 + y1y2 = 0(注意与向量共线的坐标表示)

【巩固深化,发展思维】 1.设 a = (5, 7),b = ( 6, 4),求 a?b 2.已知 A(1, 2),B(2, 3),C( 2, 5),求证:△ABC 是直角三角形. 3.教材 P114 练习 1、2 题. 4.已知 a = (3, 1),b = (1, 2),求满足 x?a = 9 与 x?b = 4 的向量 x. [展示投影]例题讲评(学生先做,学 生讲,教师提示或适当补充) 例 1. 教材 P113 例 1. 例 2. 教材 P113 例 2. [展示投影]思考 1.什么是方向向量? 2 .怎样把一个已知向量转化为单位向量? [展示投影]例题讲评(学生先做,学生讲,教师提示或适当补充) 例 3. 教材 P114 例 3. 【巩固深化,发展思维】 教材 P115 习题 A 第 1、2、3、4、5、 6 题. [学习小结] ①a = (x, y) |a|2 = x2 + y2 |a| =
? ??

x2 ? y2

( x1 ? x 2 ) 2 ? ( y1 ? y 2 ) 2 ②若 A = (x1, y1),B = (x2, y2),则| AB |=

③cos

a?b ? =| a |?|b|

x1 x 2 ? y1 y 2 x1 ? y1
2 2

x2 ? y2

2

2

④∵a b a?b = 0 即 x1x2 + y1y2 = 0 五、评价设计 1.作业:习题 2.6 B 组第 1,2,3,4 题. 2. (备选题) : ① 如图,以原点和 A(5, 2)为顶点作等腰直角△OAB,使 求点 B 和向量 AB 的坐标。 解:设 B 点坐标(x, y),则 OB = (x, y), AB = (x ∵ OB
? ??
? ??

B = 90

, B A

? ??

? ??

5, y

2) 5x

O 2y = 0

AB

∴x(x
? ??

5) + y(y

2) = 0 即:x2 + y2 5)2 + ( y

又∵| OB | = | AB |

? ??

∴x2 + y2 = (x

2)2 即:10x + 4y = 29

? 7 3 ? ? x1 ? 2 ?x 2 ? y 2 ? 5x ? 2 y ? 0 ? ? x2 ? 2 ?? 或 ? 3 ? 7 ?10x ? 4 y ? 29 ? y1 ? ? ? y2 ? ? 2 ? 2 ? 由
3 7 3 7 7 3 7 3 ?? ( ? ,? ) ( ? , ) ( ,? ) ( , ) ? ∴B 点坐标 2 2 或 2 2 ; AB = 2 2 或 2 2

②在△ABC 中, AB =(2, 3), AC =(1, k),且△ABC 的一个内角为直角, 求 k 值。
? ??

? ??

? ??

解:当 A = 90 当 B = 90

时, AB ? AC = 0,∴2×1 +3×k = 0 ∴k = 时, AB ? BC = 0, BC = AC
? ??

? ??

?

3 2
3) = ( 1, k 3)

? ??

? ??

? ??

? ??

AB = (1

2, k

∴2×(
? ??

1) +3×(k
? ??

3) = 0

11 ∴k = 3

当 C = 90 六、课后反思:

时, AC ? BC = 0,∴

1 + k(k

3) = 0

3 ? 13 2 ∴k =


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