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西瓜影视大全:高一数学必修4课件 2.4.2 平面向量数量积的坐标表示、模、夹角_图文

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1 一、复习回顾 1、数量积的定义 a ? b ? | a || b | cos? 2、向量的模 | a |? a ? a 3、向量的夹角 a?b cos ? a, b ?? | a || b | 4、向量垂直的判定 a ? b ?a?b ? 0 2 二、基础知识讲解 问题:在直角坐标系中,已知两个非零向量a ? ? x1 , y1 ?, b ? ? x2 , y2 ?,如何用向量a与b的坐标表示a b? a ? x1 i ? y1 j , b ? x2 i ? y2 j y ? a ? b ? ( x1 i ? y1 j ) ? ( x2 i ? y2 j ) 2 B ? x2 , y2 ? 2 ? x1 x2 i ? x1 y2 i ? j ? x2 y1 i ? j ? y1 y2 j b A? x1 , y1 ? j O a i ? i ? 1, j ? j ? 1 , i ? j ? j?i ? 0 ?a ? b ? x1 x2 ? y1 y2 i x 两个向量的数量积等于它们的对应坐标乘积的和. 4 二、基础知识讲解 已知非零向量a ? ? x1 , y1 ? , b ? ? x2 , y2 ? , ? a与b ?? ? 1、数量积的定义 a ? b ? | a || b | cos? a ? b ? x1 x2 ? y1 y2 随堂练习 1、已知向量a ? (1, 3), b ? ( 2, 5), 则 a?b ? 17 ;( a ? b) ? ( 2a ? b) ? 8 . 6 二、基础知识讲解 已知非零向量a ? ? x1 , y1 ? , b ? ? x2 , y2 ? ,夹角为? 1、数量积的定义 a ? b ? | a || b | cos? 2、向量的模 a ? b ? x1 x2 ? y1 y2 | a |? x ?y 2 1 2 1 | a |? a ? a 特别的,若A( x1 , y1 ), B( x2 , y2 ), 则 | AB |? ( x2 ? x1 ) 2 ? ( y2 ? y1 ) 2 随堂练习 2、已知向量AB ? (1, 3), AC ? (2, 5), 则 | BC |? 5 7 ; 二、基础知识讲解 已知非零向量a ? ? x1 , y1 ? , b ? ? x2 , y2 ? , ? a与b ?? ? 1、数量积的定义 a ? b ? | a || b | cos? 2、向量的模 a ? b ? x1 x2 ? y1 y2 | a |? x ?y 2 1 2 2 1 | a |? a ? a 3、向量的夹角 特别的,若A( x1 , y1 ), B( x2 , y2 ), 则 | AB |? ( x2 ? x1 ) ? ( y2 ? y1 ) 2 a?b cos ? a, b ?? | a || b | cos? ? x1 x2 ? y1 y2 x12 ? y12 x22 ? y22 8 二、基础知识讲解 3、向量的夹角 a?b cos ? a, b ?? | a || b | cos? ? x1 x2 ? y1 y2 x ?y 2 1 2 1 x2 ? y2 2 2 随堂练习 ? 夹角为 4 3、已知向量a ? (1, 1), 2a ? b ? ( 4, 2), 则a与b的 ; 9 三、例题分析 例1、已知?AOB中,O为原点,A( 2, 2), B(? , 1) 且?ABO是钝角,求?的取值范围 分析: ?ABO是钝角 ? BO ? BA ? 0且不共线 11 二、基础知识讲解 已知非零向量a ? ? x1 , y1 ? , b ? ? x2 , y2 ? , ? a与b ?? ? 1、数量积的定义 a ? b ? | a || b | cos? 2、向量的模 a ? b ? x1 x2 ? y1 y2 | a |? x12 ? y12 | a |? a ? a 3、向量的夹角 a?b cos ? a, b ?? | a || b | 4、向量垂直的判定 cos? ? x1 y1 ? x2 y2 x12 ? y12 x22 ? y22 12 a ? b ?a?b ? 0 a ? b ? x1 y1 ? x2 y2 ? 0 三、例题分析 例2、已知A(1、2),B(2,3),C(2,5),试判断 ΔABC的形状,并给出证明。 y C 证明: AB ? ? 2 ? 1, 3 ? 2 ? ? ?1,1? AC ? ? ?2 ? 1, 5 ? 2 ? ? ? ?3, 3 ? AB AC ? 1? ? ?3? ? 1? 3 ? 0 A B ? AB ? AC ∴ΔABC是直角三角形 O x 注:两个向量的数量积是否为零是判断相应的两条直 线是否垂直的重要方法之一。 如证明四边形是矩形,三角形的高,菱形对角线 13 垂直等。 三、例题分析 例2、已知A(1、2),B(2,3),C(2,5),试判断 ΔABC的形状,并给出证明。 变式:在Rt?ABC中, AB ? (2, 3), AC ? (1, k), 求k的值 分析:由?ABC为直角三角形,需对内角进行分类讨论 若A ? 90o,则AB ? AC ? 0 若B ? 90o,则BA ? BC ? 0 若C ? 90o,则CA ? CB ? 0 14 三、例题分析 例3、已知?ABC的顶点坐标为A ? 2, ?1? ,B ? 3, 2 ? , C ? ?3, ?1? ,BC边上的高为AD,求点D及 AD的 坐标及?ABC的面积。 15 四、针对性练习 1、已知a ? ( ?2, 4), b ? (1, ?2),则a与b的关系是 ? A、不共线且不垂直 C、共线同向 B、垂直 D、共线反向 D? 2、以A (2,5),B (5,

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