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c31b.vip:人教A版高中数学必修四 2.4.2平面向量数量积的坐标表示 名师公开课市级获奖课件(11张)

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2.4.2平面向量数量积的 坐标表示、模、夹角 复习引入 1. 平面向量的数量积(内积)的定义: 已知两个非零向量 a和b, 它们的 夹角为? ,我们把数量| a || b | cos ?叫 做 a 与 b 的数量积(或内积) . 记为: a ? b , 即 a ? b ? | a || b | cos? . 规定: 零向量与任一向量的数 量积 为0,即a ? 0 ? 0 . 2. 两个向量的数量积的性质: 设 a、 b为两个非零向量 , e 是与 b同向的单位向量. (1) e ? a ? a ? e ? a ? cos ? . (2) a ? b ? a ? b ? 0 . ( 3) 当 a 与 b 同向时, a ? b ? a ? b . 当 a 与 b 反向时, a ? b ? ? a ? b . 特别地, a ? a ? a 或 a ? a ? a . (4) cos ? ? a?b a b . ( 5) a ? b ? a b . 2 复习引入 3. 练习: (1) 已知 a ? 1, b ? 2 , 且(a ? b)与a垂直, 则 a 与 b 的夹角是( ) A. 60 o B. 30 o C. 135 o D. 45 o ( 2) 已知 a ? 2, b ? 1, a 与 b 之间的夹角 为 , 那么向量 m ? a ? 4b 的模为 ( 3 ? ) A. 2 B. 2 3 C. 6 D. 12 讲授新课 1. 平面两向量数量积的坐标表示: 探究: 已知两个非零向量 a ? ( x1 , y1 ), b ? ( x 2 , y2 ), 怎样用 a 和 b 的坐标 表示 a ? b ? 两个向量的数量积等于它们对应坐标 的乘积的和. 即 a ? b ? x1 x2 ? y1 y2 . 2.平面内两点间的距离公式: (1) 设 a ? ( x , y ), 则 a ?x ?y或 a? 2 2 2 x ?y . 2 2 (2)如果表示向量 a 的有向线段的起点 和终边的坐标分别为( x1 , y1 ), ( x2 , y2 ), | a |? ( x1 ? x2 ) ? ( y1 ? y2 ) 2 2 (平面内两点间的距离公式) 3.向量垂直的判定: 设a ? ( x1 , y1 ), b ? ( x2 , y2 ), 则 a ? b ? x1 x2 ? y1 y2 ? 0. 4.两向量夹角的余弦: 讲解范例: 例1. 已知A(1,2),B(2,3),C(?2,5), 试判断△ABC的形状,并给出证明. 例2. 设 a ? (5, ?7), b ? ( ?6, ?4), 求 a ? b 及 a、b 间的夹角? (精确到1 ). o 例3. 已知 a ? (1, 3), b ? ( 3 ? 1, 3 ? 1), 则 a 与 b 的夹角是多少 ? 评述:已知三角形函数值求角时,应注重角 的范围的确定. 练习: 1.教材P.107练习第1、2、3题. 2. 已知A(3,2),B(-1,-1),若点 1 P ( x , ? )在线段AB的中垂线上,则 2 1 2 x= . 课堂小结 1. a ? b ? x1 x2 ? y1 y2 . 2. 平面内两点间的距离公式: | a |? ( x1 ? x2 ) ? ( y1 ? y2 ) 2 2 3. 向量垂直的判定: a ? b ? x1 x2 ? y1 y2 ? 0. 课后思考: 1. 以原点和A(5, 2)为顶点作等腰直角 △OAB,使?B=90?,求点B和向量 的坐标. 2. 在△ABC中, 且△ABC的一个内角为直角,求k值.

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