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lifei: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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一、复习回顾 ? ? ? ? a ? b ? | a || b | cos? 2、向量的模 ? ? ? | a |? a ? a 1、数量积的定义 ? ? ? ? a?b cos ? a, b ?? ? ? | a || b | 4、向量垂直的判定 3、向量的夹角 ? ? ? ? a ? b ?a?b ? 0 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? y ? ? ? ? ? ? ? a ? x1 i ? y1 j , b ? x2 i ? y2 j ? ? ? ? ? ? B ? x2 , y2 ? ? ? a ? b ? ( x1 i ? y1 j ) ? ( x2 i ? y2 j ) ?2 ? ? ? ? ?2 b ? x1 x2 i ? x1 y2 i ? j ? x2 y1 i ? j ? y1 y2 j ? j ? ? ? ? ? ? ? ? ? i ? i ? 1, j ? j ? 1 , i ? j ? j?i ? 0 ? ? ? O i ?a ? b ? x1 x2 ? y1 y2 ? a A? x1 , y1 ? x 两个向量的数量积等于它们的对应坐标乘积的和. 二、基础知识讲解 ? ? ? ? 已知非零向量a ? ? x1 , y1 ? , b ? ? x2 , y2 ? , ? a与b ?? ? ? ? ? ? a ? b ? | a || b | cos? 1、数量积的定义 ? ? a ? b ? x1 x2 ? y1 y2 ? ? 1、已知向量a ? (1, 3), b ? ( 2, 5), 则 ? ? ? ? ? ? 17 a?b ? ;( a ? b) ? ( 2a ? b) ? 随堂练习 8 . 二、基础知识讲解 ? ? 已知非零向量a ? ? x1 , y1 ? , b ? ? x2 , y2 ? ,夹角为? ? ? ? ? a ? b ? | a || b | cos? 2、向量的模 ? ? ? | a |? a ? a 1、数量积的定义 ? ? a ? b ? x1 x2 ? y1 y2 ? 2 2 | a |? x1 ? y1 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 ; 二、基础知识讲解 ? ? ? ? 已知非零向量a ? ? x1 , y1 ? , b ? ? x2 , y2 ? , ? a与b ?? ? ? ? ? ? a ? b ? | a || b | cos? 2、向量的模 ? ? ? | a |? a ? a 1、数量积的定义 ? ? a ? b ? x1 x2 ? y1 y2 ? 2 2 | a |? x1 ? y1 特别的,若A( x1 , y1 ), B( x2 , y2 ), 则 ??? ? 2 2 | AB |? ( x2 ? x1 ) ? ( y2 ? y1 ) ? ? ? ? a?b cos ? a, b ?? ? ? | a || b | 3、向量的夹角 cos? ? x1 x2 ? y1 y2 x12 ? y12 x22 ? y22 二、基础知识讲解 ? ? ? ? a?b cos ? a, b ?? ? ? | a || b | 3、向量的夹角 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 ; 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 二、基础知识讲解 ? ? ? ? 已知非零向量a ? ? x1 , y1 ? , b ? ? x2 , y2 ? , ? a与b ?? ? ? ? ? ? a ? b ? | a || b | cos? 2、向量的模 ? ? ? | a |? a ? a 1、数量积的定义 ? ? a ? b ? x1 x2 ? y1 y2 ? | a |? x12 ? y12 ? ? ? ? a?b cos ? a, b ?? ? ? | a || b | 4、向量垂直的判定 3、向量的夹角 cos? ? x1 y1 ? x2 y2 x12 ? y12 x22 ? y22 ? ? ? ? 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 ? B ???? ???? ? AB ?AC ? 1? ? ?3? ? 1? 3 ? 0 A ???? ???? x O ? AB ? AC ∴ΔABC是直角三角形 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,

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