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www.www609000.com: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、模、夹角》教案

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第 9 课时 三、平面向量数量积的坐标表示、模、夹角 教学目的: ⑴要求学生掌握平面向量数量积的坐标表示 ⑵掌握向量垂直的坐标表示的充要条件,及平面内两点间的距离公式. ⑶能用所学知识解决有关综合问题. 教学重点:平面向量数量积的坐标表示 教学难点:平面向量数量积的坐标表示的综合运用 授课类型:新授课 教 具:多媒体、实物投影仪 教学过程: 一、复习引入: 1.两个非零向量夹角的概念 已知非零向量a与b,作 OA =a, OB =b,则∠AOB=θ (0≤θ ≤π )叫a与b的夹 角. 2.平面向量数量积(内积)的定义:已知两个非零向量a与b,它们的夹角是θ ,则数量 |a||b|cos ?叫a与b的数量积,记作 a?b,即有 a?b = |a||b|cos ?, (0≤θ ≤π ).并规定 0 与任何向量的数量积为 0. 3.向量的数量积的几何意义: 数量积 a?b 等于 a 的长度与 b 在 a 方向上投影|b|cos ?的乘积. 4.两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? 3? C e?a = a?e =|a|cos ?; 2? a ? b ? a ?b = 0 2 当 a 与 b 同向时,a?b = |a||b|;当 a 与 b 反向时,a?b = ?|a||b|. 特别的 a?a = |a| 或 | a |? a ? a 4? cos ? = a ?b ;5 ?|a?b| ≤ |a||b| | a || b | 5.平面向量数量积的运算律 交换律:a ? b = b ? a 数乘结合律:( ? a)?b = ? (a?b) = a?( ? b) 分配律:(a + b)?c = a?c + b?c 二、讲解新课: ⒈ 平面两向量数量积的坐标表示 已知两个非零向量 a ? ( x1 , y1 ) , b ? ( x2 , y 2 ) ,试用 a 和 b 的坐标表示 a ? b . 设 i 是 x 轴上的单位向量, j 是 y 轴上的单位向量,那么 a ? 所以 a ? b ? ( x1i ? x1i ? y1 j , b ? x2 i ? y2 j y1 j )(x2 i ? y2 j ) ? x1 x2i 2 ? x1 y2i ? j ? x2 y1i ? j ? y1 y2 j 2 ? x1 x2 ? y1 y2 ? x1 x2 ? y1 y2 又 i ? i ? 1 , j ? j ? 1 , i ? j ? j ? i ? 0 ,所以 a ? b 这就是说:两个向量的数量积等于它们对应坐标的乘积的和.即 a ? b 2. 平面内两点间的距离公式 一、 设 a ? ( x, y) ,则 | a | 2 ? x 2 ? y 2 或 | a |? x 2 ? y 2 . (2)如果表示向量 a 的有向线段的起点和终点的坐标分别为 ( x1 , y1 ) 、 ( x2 , y 2 ) ,那么 | a |? ( x1 ? x 2 ) 2 ? ( y1 ? y 2 ) 2 (平面内两点间的距离公式) 二、 向量垂直的判定 设a ? ( x1 , y1 ) , b ? ( x2 , y 2 ) ,则 a ? b a ?b ? | a |?| b | x1 x 2 ? y1 y 2 x1 ? y1 2 2 ? x1 x2 ? y1 y2 ? 0 三、 两向量夹角的余弦( 0 ? ? ? ? ) cos? = x2 ? y2 2 2 四、 讲解范例: 五、 设 a = (5, ?7),b = (?6, ?4),求 a·b 及 a、b 间的夹角θ (精确到 1 ) o 例 2 已知 A(1, 2),B(2, 3),C(?2, 5),试判断△ABC 的形状,并给出证明. 例 3 已知 a = (3, ?1),b = (1, 2),求满足 x?a = 9 与 x?b = ?4 的向量 x. 解:设 x = (t, s), 由 x?a ? 9 ? 3t ? s ? 9 ?t?2 ?? ?? x ? b ? ?4 ?t ? 2s ? ?4 ?s ? ?3 ∴x = (2, ?3) 例 4 已知 a=(1, 3 ),b=( 3 +1, 3 -1),则 a 与 b 的夹角是多少? 分析:为求 a 与 b 夹角,需先求 a·b 及|a|·|b|,再结合夹角 θ 的范围确定其值. 解:由 a=(1, 有 a·b= 3 ),b=( 3 +1, 3 -1) 3 +1+ 3 ( 3 -1)=4,|a|=2,|b|=2 2 . a ?b 2 ? a?b 2 记 a 与 b 的夹角为 θ ,则cosθ = 又∵0≤θ ≤π ,∴θ = ? 4 评述:已知三角形函数值求角时,应注重角的范围的确定. 例 5 如图,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 AB 的坐标. 解:设 B 点坐标(x, y),则 OB = (x, y), AB = (x?5, y?2) ∵ OB ? AB ∴x(x?5) + y(y?2) = 0 即:x + y ?5x ? 2y = 0 2 2 又∵| OB | = | AB | ∴x + y = (x?5) + (y?2) 即:10x + 4y = 29 2 2 2 2 ? 7 3 ? x1 ? x2 ? ?x 2 ? y 2 ? 5x ? 2 y ? 0 ? ? ? 2 或? 2 ?? 由? 3 7 10 x ? 4 y ? 29 ? ? y1 ? ? ? y2 ? ? 2 ? 2 ? ∴B 点坐标 ( ,? ) 或 ( , ) ; AB = ( ? 7 2 3 2 3 7 2 2 3 7 7 3 ,? ) 或 ( ? , ) 2 2 2 2 例 6 在△ABC 中, AB =(2, 3), AC =(1, k),且△A

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