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seye.88.com:人教版高中数学必修4-2.4《平面向量数量积的坐标表示、模、夹角》参考教案1

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§2.4 平面向量的数量积 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? 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 的 a?a = |a|2 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 , y2 ) ,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 ? x1i ? y1 j , b ? x2i ? y2 j 所以 a ? b ? (x1i ? y1 j)( x2i ? y2 j) ? x1x2i 2 ? x1 y2i ? j ? x2 y1i ? j ? y1 y2 j 2 又 i ? i ? 1 , j ? j ? 1, i ? j ? j ? i ? 0,所以 a ? b ? x1x2 ? y1 y2 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 ? x1x2 ? y1 y2 2. 平面内两点间的距离公式 一、设 a ? (x, y) ,则| a |2 ? x2 ? y 2 或| a |? x2 ? 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 , y2 ) , 那么| a |? (x1 ? x2 )2 ? ( y1 ? y2 )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 , y2 ) ,则 a ? b ? x1x2 ? y1 y2 ? 0 三、两向量夹角的余弦( 0 ? ? ? ? ) cos? = a ? b ? x1 x2 ? y1 y2 | a |?| b | x12 ? y12 x2 2 ? y2 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 1o) 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 x ? b ? ?4 ? ? 3t ??t ? ?s ?9 2s ? ?4 ? ?t ??s ?2 ? ?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, 3 ),b=( 3 +1, 3 -1) 有 a·b= 3 +1+ 3 ( 3 -1)=4,|a|=2,|b|=2 2 . 记 a 与 b 的夹角为 θ ,则cosθ = a ? b ? 2 a?b 2 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:x2 + y2 ?5x ? 2y = 0 又∵| OB | = | AB | ∴x2 + y2 = (x?5)2 + (y?2)2 即:10x + 4y = 29 由 ? ? x 2 ? y2 ? 5x ? ?10x ? 4 y ? 29 2y ? 0 ? ? ??x1 ? ??? y1 ? ? 7 2 ? 3 2 ? 或 ?? ? ? x2 y2 ? ? 3 2 7 2 ∴B 点坐标 (7 ,? 3) 或 ( 3 , 7) ; AB = (? 3 ,? 7) 或 (? 7 , 3) 2 2 22 22 22 例 6 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 =

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