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? § 2.4.2 平面向量数量积的 模、夹角 坐标表示、 引入 练1、已知 | a |? 6, | b |? 4, a与b的夹角为 600 , 求(a ? 2b) ? (a ? 3b). 练 2、已知 | a |? 3, | b |? 4, 且a与b不共线.k为 何值时, 向量a ? k b与a ? k b垂直. 练 3、已知 | a |? 2, | b |? 5, a ? b ? ?3, 求 | a ? b |, |a?b|. 新课 数量积的坐标表示 : 已知两个非零向量 a ? ( x1 , y1 ), b ? ( x 2 , y2 ), 则a ? b ? x1 x2 ? y1 y2 . (1)若a ? ( x, y ), 则 | a |2 ? ____, 或 | a |? ____; ( 2)若A( x1 , y1 ), B( x2 , y2 )则 | AB |? ____; ( 3)若a ? ( x1 , y1 ), b ? ( x 2 , y2 ), 则a ? b ? __ __ . 举例 例1、求证 : A(1,0), B( 5,?2), C ( 8,4), D(4,6) 为顶点的四边形是一个 矩形. 解: | AB |? ( 4, ?2 ) ? DC , cos ? AB , AD ? ? 其余略;所以得证 ( 4 ,?2 )( 3 ,6 ) 4 2 ? ( ? 2 )2 32 ? 62 向量夹角公式: cos ? ? a?b | a || b | ? x1 x 2 ? y1 y2 x1 ? y1 2 2 x2 ? y2 2 2 例2、已知 | a |? 3 , b ? ( 1 ,2 ), 且a // b , 求a的坐 标. x y 解: ? 平 行? 设 a ? (x , y) ,有 ? ,且 1 2 x ? y 2 2 ? 9, 所 以 5x 6 5, 5 5) 2 ? 9, x ? 3 5 5 ,y? 6 5 5 3 a ?( 5 例3、已知a ? ( 4 ,2 ), 求与a垂直的单位向量 的坐标. 解:设单位向量为 i ? ( x , y ),有a ? i ? 0 5 2 5 ( x , y )( 4 ,2 ) ? 0 , 且x ? y ? 1 , 解 得i ? ( ,? ) 5 5 2 2 5 2 5 或( ? , ) 5 5 小结 数量积的坐标表示 : 已知两个非零向量 a ? ( x1 , y1 ), b ? ( x 2 , y2 ), 则a ? b ? x1 x2 ? y1 y2 . 向量夹角公式: cos ? ? a?b | a || b | ? x1 x 2 ? y1 y2 x1 ? y1 2 2 x2 ? y2 2 2 作业 课本107页练习 谢谢观看!

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