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北师大版数学精品教学资料 平面向量数量积的坐标表示 教学目标 1.正确理解掌握两个向量数量积的坐标表示方法,能通过两个向量的坐标 求出这两个向量的数量积. 2.掌握两个向量垂直的坐标条件,能运用这一条件去判断两个向量垂直. 3.能运用两个向量的数量积的坐标表示去解决处理有关长度、角度、垂直 等问题. 重点:两个向量数量积的坐标表示,向量的长度公式,两个向量垂直的充 要条件. 难点:对向量的长度公式,两个向量垂直的充要条件的灵活运用. 教学过程设计 (一)学生复习思考,教师指导. 1.A 点坐标(x1,y1),B 点坐标(x2,y2). =________ =________ 2.A 点坐标(x1,y1),B 点坐标(x2,y2) =________ 3.向量的数量积满足那些运算律? (二)教师讲述新课. 前面我们已经学过了两个向量的数量积,如果已知两个向量的坐标, 如何用这些坐标来表示两个向量的数量积,这是一个很有价值的问题. 设两个非零向量为 =(x1,y1), =(x2,y2). 为 x 轴上的单位向 量, 为 y 轴上的单位向量,则 =x1 +y1 , =x2 +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. 引入向量的数量积的坐标表示,我们得到下面一些重要结论: (1)向量模的坐标表示: (2)平面上两点间的距离公式: 向量 的起点和终点坐标分别为 A(x1,y1),B(x2,y2), = (3)两向量的夹角公式 设 =(x1,y1), =(x2,y2), =θ. 4.两向量垂直的充要条件的坐标表示 =(x1,y1), =(x2,y2). 即两向量垂直的充要条件是它们对应坐标乘积的和为零. (三)学生练习,教师指导. 练习 1:课本练习 1. 已知 a(-3,4), (5,2) 练习 2:课本练习 2. 已知 =(2,3), =(-2,4), =(-1,-2). · =2×(-2)+3×4=8,( + )·( - )=-7. ·( + )=0,(a+b)2=(0,7)·(0,7)=49. 练习 3:已知 A(1,2),B(2,3),C(-2,5). 求证:△ABC 是直角三角形. 证:∵ =(1,1), =(-3,3), =(-4,2). 经检验, · =1×(-3)+1×3=0. ∴ ⊥ ,△ABC 是直角三角形. (四)师生共同研究例题. 例 1:已知向量 =(3,4), =(2,-1). (1)求 与 的夹角 θ, (2)若 +x 与 - 垂直,求实数 x 的值. 解:(1) =(3,4), =(2,-1). (2) +x 与 - 垂直, ( +x )·( - )=0, +x =(3,4)+x(2,-1)=(2x+3,4-x) - =(3,4)-(2,-1)=(1,5). 例 2:求证:三角形的三条高线交于一点. 证:设△ABC 的 BC、AC 边上的高交于 P 点,现分别以 BC、PA 所在直线 为 x 轴、y 轴,建立直角坐标系,设有关各点的坐标为 B(x1,0),C(x2,0), A(0,y1),P(0,y). ∵ ⊥ , =(-x1,y), (-x1)×(-x2)+y×y1=0. 即 x1x2+yy1=0. =(-x2,y1). 又 =(-x2,y), =(-x1,y1). · =(-x1)×(-x2)+y×y1=x1x2+yy1=0. ∴ ⊥ ,CP 是 AB 边上的高. 故三角形的三条高线交于一点. (五)作业.习题 5.7 1,2,3,4,5.

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