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WWW,41616,com:新人教A版必修4高中数学2.3.4平面向量共线的坐标表示导学案

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高中数学 2.3.4 平面向量共线的坐标表示导学案 新人教 A 版必修 4 学习目标 1、在理解向量共线的概念的基础上,学习用坐标表示向量共 线的条件。 2、利用向量共线的坐标表示解决有关问题。 学习过程 一、课前准备(预习教材 P98—P100) 复习: ⑴若点 A 、 B 的坐标分别为 ? x1 , y1 ? , ? x2 , y2 ? 那么向量 AB 的坐标 为 . ,a ?b ? ⑵若 a ? ? x1 , y1 ? , b ? ? x2 , y2 ? ,则 a ? b ? , ?a ? 二、新课导学 ※ 探索新知 探究 :平面向量共线的坐标表示 1 问题 1:两向量平行(共线)的条件是什么? 若 a, b ( b ? 0 )共线,当且仅当存在实数 ? ,使 。 问题 2:假设 a ? ? x1 , y1 ? , b ? ? x2 , y2 ? ( b ? 0 ) ,用坐标该如何表示这 两个向量共线呢? 2、设 a ? ( x1, y1 ), b ? ( x2 , y2 ) ,其中 b ? 0 ,则 a // b 等价于 ______________________。 ※ 典型例题 例 1、已知 a ? ?4,?2?, b ? ? 6, y ? ,且 a // b ,求 y . 2 例 2、向量 OA ? ? k,12? , OB ? ? 4,5? , OC ? ?10, k ? ,当 k 为何值时, A, B, C 三点共线. 变式 :证明下列各组点共线: B(?3, ?4)、C (2, ) (1) A(1, 2),、 Q(1, ?3)、R(8, ) (2) P(9,1) 、 1 2 7 2 例 3、设点 P 是线段 P1 P2 上的一点, P1 , P2 的坐标分别是 ? x1 , y1 ? , ? x2 , y2 ? . ⑴当点 P 是线段 P1 P2 的中点时,求点 P 的坐标; ⑵当点 P 是线段 P1 P2 的一个三等分点时,求点 P 的坐标 . 3 *变式: 当 PP ? ? PP2 ,点 P 的坐标是什么? 1 三、小结反思 ※ 当堂检测(时量:5 分钟 满分:10 分)计分: 1 已知 A(?2, ?3), B(2,1),C(1, 4),D(?7, ?4) 判断 AB 与 CD 是否共线? 2、已知 a ? ? 2, ?1? , b ? ? x,2? , c ? ? ?3, y ? ,且 a // b // c ,求 x, y 的值. 4 3、平面内给定三个向量 a =(3,2), b =(-1,2), c =(4,1), 求: (1)求 3 a + b -2 c ; (2)求满足 a =m b +n c 的实数 m,n; (3)若( a +k c ) / / (2 b - a ),求实数 k. 课后作业 1. 已知 a ? ?1,2 ? , b ? ? x,1? ,若 a ? 2b 与 2a ? b 平行,则 x 的值 为 . 2、已知 O 是平面上一定点,A、B、C 是平面上不共线的三点, →+λ ( → 动点 P 满足→ OP=OA AB+→ AC),λ ∈[0,+∞),则点 P 的 轨 迹一定通过△ABC 的( ) 5 A.外心 B.垂心 C.内心 D.重心 3、已知四点 A(x,0)、B(2x,1)、C(2,x)、D(6,2x). (1)求实数 x,使两向量→ AB、→ CD共线. →与→ (2)当两向量AB CD共线时,A、B、C、D 四点是否在同一条 直线上? 6

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