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yabovip9753com:2019—2020年最新北师大版高中数学必修四2.6平面向量数量积的坐标表示教案(精品教学设计)

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2.6 平面向量数量积的坐标表示 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. 美好的未来不是等待,而是孜孜不倦的攀登! 难点: 用坐标法处理长度、角度、垂直问题. 三.学法与教法 (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. 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 ? 美好的未来不是等待,而是孜孜不倦的攀登! |a| = x2 ? y2 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 ? ?? ( x1 ? x 2 ) 2 ? ( y1 ? y 2 ) 2 x2 ? y2 2 2 ④∵a?b ? 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. 美好的未来不是等待,而是孜孜不倦的攀登! 【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 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 ( x1 ? x 2 ) 2 ? ( y1 ? y 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, B 使?B = 90?, 求点 B 和向量 AB 的坐标。 ? ?? ? ?? A O 解:设 B 点坐标(x, y),则 OB = (x, y), AB = (x?5, y?2) ∵ OB ? AB 2y = 0 又∵| OB | = | AB | 10x + 4y = 29 ? 7 3 ? x1 ? x2 ? ?x 2 ? y 2 ? 5x ? 2 y ? 0 ? ? ? 2 或? 2 ?? 由? 3 7 ?10x ? 4 y ? 29 ? y1 ? ? ? y2 ? ? 2 ? 2 ? ? ?? ? ?? ? ?? ? ?? ∴x(x?5) + y(y?2) = 0 即:x2 + y2 ?5x ? ∴x2 + y2 = (x?5)2 + (y?2)2 即: ∴B 点坐标 ( 7 ,? 3 ) 或 ( 3 , 7 ) ; AB = (? 3 ,? 7 ) 或 (? 7 , 3 ) 2 2 2 2 2 2 2 2 美好的未来不是等待,而是孜孜不倦的攀登! ? ?? 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?时,AB ? AC = 0, ∴2×1 +3×k = 0 k =? 3 2 ? ?? ? ?? ? ?? ? ?? ∴ AB ? BC = 0, BC = AC ? AB = (1?2, k?3) 当 B = 90?时, ? ?? ? ?? ? ?? ? ?? ? ?? = (?1, k?3) ∴ 2× (?1) +3 × (k?3) = 0 = 11 3 ∴k AC ? BC = 0, 当 C = 90?时, ∴?1 + k(k?3) =

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