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www.001dd.com:人教版高中数学必修4精品课件2.4.2 平面向量数量积的坐标表示、模、夹角_图文

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2.4.2 平面向量数量积的 坐标表示、模、夹角 一、复习引入 1、数量积的定义: a ? b ?| a || b | cos ? | b | cos ? 叫做 b 在 a方向上的投影 2、投影: B b O B1 | b | cos ? θ a A 3.数量积的性质 (1)a ? b ? a ? b ? _______ . 0 证明向量 垂直的依据 | a || b | ; (2)若 a 与 b 同向,a ? b ? _______ | a | . | a |? a ? a ? _____ ≤ 若 a 与 b 反向,a ? b ? _______ ? | a || b | ; 2 a?a (3) | a ? b | ____ | a || b | .( 填 ? 或 ?) (4) cos? ? _________ a b a ?b 我们学过两向量的和与差可以转 化为它们相应的坐标来运算,那么怎 样用 a和b的坐标表示 a ? b呢? 二、探究解疑 1、平面向量数量积的坐标表示 i 是x轴上的单位向量, 问题1、如图, 是y轴上的单位向量, j y A(x1,y 1 j ? j ?1 i ?i ? . . ) 1 B(x2,y a j i ? j ? j ? i ?0 . 2) b o i x 问题2 已知两非零向量 a? (x1,y1 ), b? (x2,y2) 设, i j分别为与x轴和y轴方向相同的单位向量,则 a ? x1 i ? j表示? b ? x2 i ? y2 j a、 b如何用与 i y1 j ?a ? b ? (x1 i ? y1 j ) ( ? x 2 i ? y2 j ) ? x1 x2 ? y1 y2 ? x1 x2 i ? x1 y2 i ? j ? x2 y1 j ? i ? y1 y2 j ? a ? b ? x1 x 2 ? y1 y2 2 2 两个向量的数量积等于它们对应坐标的乘积的和。 例1 设a = (5, ?7),b = (?6, ?4), ? ? ? ? 求a ·b及|a|的值 ? a b=5 ? ? -6 ? + ? -7 ? ? ? -4 ? =-2 a = a a = ?5,-7? ?5,-7? =5 ? 5+ ? -7? ? ?-7? =74 2 a = 74 思考:在求 a 问题中,能否推广到一般情况? 2.向量的长度(模) 设a =(x,y),则 |a|2= x ?y x ? y 或|a |= _______ 2 2 2 2 2 2 思考:知道向量的起点和终点坐标,如何表示 a ? x2 ? x1 ? ? ? y2 ? y1 ? 若设A(x1,y1)、B(x2,y2),则 |AB|= ___________ 平面内两点间的距离公式 3、向量平行和垂直的坐标表示式 设两个非零向量 a ? ( x1 , y1 ), b ? ( x2 , y2 ), 则 a // b ? x1 y2 ? x2 y1 ? 0 a⊥b a· b=0 x1x2+y1y2=0 a ? b ? x1x2 ? y1 y2 ? 0 4、两向量夹角余弦的坐标运算 两 非零 向量 a? (x1,y1 ), b? (x2,y2) ,夹 角为 ? a?b cos ? ? | a || b | ? x1 x2 ? y1 y2 x ?y 2 1 2 1 x ?y 2 2 2 2 例题2、已知a = 1, 3 ,b= ? ? ? ? 2 3+1, 3-1 ,求a与b的夹角 ? 设a与b的夹角为?, cos ? = ab a b = ? 1 +? 3? ? 1? 2 3+1 + 3 ? ? 2 3-1 ? ? 2 3+1 + ? ? 3-1 2 = 2 ? =45 变式:已知a= ? x,-2? ,b= ? 3,5? ,且向量a与b 的夹角是钝角, 求x的取值范围。 三、典例分析 例3已知A(1,2),B(2,3),C(-2,5), 试判断?ABC的形状,并给出证明. y 证明: 向量的数量 C(-2,5) ? ? ? ? ? AB ? 2 ? 1 , 3 ? 2 ? 1 , 1 方法1 积是否为零, AC ? ? ?2 ? 1,5 ? 2? ? ? ?3,3? 是判断相应 B(2,3) ? AB? AC ? 1? ?? 3? ?1? 3 ? 0 的两条线段 或直线是否 A(1,2) ? AB ? AC 垂直的重要 x ∴ △ABC是直角三角形 0 方法之一 方法2: AB = ?1,1?, AC = ? -3,3?, BC = ? -4,2 ? | AB |? 2,|AC |? 3 3,|BC |? 2 5 | AB | ?|AC | ?|BC | ,?ABC是直角三角形 2 2 2 变式练习:已知?ABC是直角三角形, AC ? ? 3,2 ?, BC = ? k ,1?,求k的值。 11 2 3 ? 13 k = ,k =- ,k = 3 3 2 例题4、已知a = ?1,-2 ? ,b= ? 2,0 ? ,求同时满足 a c =4,b c =0的向量c。 设c = ? x,y ? ? ?a c =4 ?x-2y = 4 由? 得? , ? x =0,y =-2 ? ?b c =0 ?2x=0 c = ? 0,-2 ? 四、反思提高 ? ? ? ? 1、若 a ? (?3,4),b ? (5,12),则a 与 b 夹角的余弦值为 33 . 65 2、已知a ? (?3,4), b ? (5,2),则 a ? 5 ___, b ? ____, 29 a ? b ? ______ ?7 . 四、反思提高 3、 1/2 4、 23 . 3 小结: 1.平面向量数量积的坐标表示. 2.判断两个向量垂直的方法. 3.平面

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