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www.ag2641.com:高中数学 2.4.2平面向量数量积的坐标表示模夹角课件 新人教版必修4_图文

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2 .4 .2 平面向量数量积的 坐 标表示﹑模﹑夹角 北 A 45° 东 B 复习引入 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? 新课学习 1、平面向量数量积的坐标表示 j y轴 如图,是 i x轴上的单位向量,是 上的单位向量, y 因为 a ? b ? a ? b cos ? 所以 1 1 j ? j ? . . i ?i ? i ? j ? j ?i ? 0 . j o i x 下面研究怎样用 a和b的坐标表示 a ? b. 设两个非零向量 a = (x1 , y1 ), b = (x2 , y2 ) 则 a ? x1 i ? y1 j 2 b ? x2 i ? y2 j , b y A(x1,y1) a i a ? b ? ( x1 i ? y1 j ) ? ( x2 i ? y2 j ) B(x2,y2) ? x1 x2 i ? x1 y2 i ? j ? x2 y1 i ? j ? y1 y2 j 2 j o x ? x1 x2 ? y1 y2 故两个向量的数量积等于它们 对应坐标的乘积的和。即 y A(x ,y ) 1 1 a ? b ? x1x2 ? y1 y2 B(x2,y2) b j a i o 根据平面向量数量积的 x 坐标表示,向量的数量积的运算可 转化为向量的坐标运算。 2、向量的模和两点间的距离公式 (1) a ? a ? a 或 a ? 2 a ? a; (1)向量的模 设a ? ( x, y ), 则 a ? x ? y , 或 a ? 2 2 2 2 2 x ?y ; (2)两点间的距离公式 设A(x1 , y1 )、B ( x2 , y2 ), 则 AB ? (x1 ? x2 ) ? (y1 ? y2 ) 2 2 3、两向量垂直和平行的坐标表示 (1)垂直 a ? b ? a ? b ? 0 设a ? (x1 , y1 ), b ? ( x2 , y2 ), 则 a ? b ? x1 x2 ? y1 y2 ? 0 (2)平行 设a ? (x1 , y1 ), b ? ( x2 , y2 ), 则 a// b ? x1 y2 ? x2 y1 ? 0 4、两向量夹角公式的坐标运算 设a与b 的夹角为?(0 ? ? ? 180 ), ? ? 则 cos ? ? a ?b ab 设a ? (x1 , y1 ), b ?( x2 , y2 ), 且a与b夹角为?, (0 ? ? ? 180 )则 cos ? ? ? ? 2 1 2 1 2 2 x1 x2 ? y1 y2 x ?y ? x ?y 2 1 2 1 2 2 2 2 2 2 . 其中 x ? y ? 0, x ? y ? 0. 三、基本技能的形成与巩固 例1 (1)已知a ? (?1,2 ? 3 ), b ? (1,1), 求a ? b, a ? b, a与b的夹角? . 解: a ? b ? 1 ? 3,  a ? b ? 2 4 ? 2 3 ? 2(1 ? 3) 1     cos ? ? ? ,   0 ? ? ? 180 ,   ?? ? 60 . a?b 2 a ?b (2)已知a ? (2,3), b ? (?2,4), 则(a ? b) ( ? a ? b) ? . 法一: a ? b ? (0,7), a ? b ? (4,?1) ? (a ? b) ( ? a ? b) ? 0 ? 4 ? 7 ? (?1) ? ?7. 法二:(a ? b) ( ? a ? b) ? a ?b 2 2 2 2 ? a ? b ? 13 ? 20 ? ?7 练习1:课本P120 1、2. 例2 已知A(1,2),B(2,3),C(-2,5), 试判断?ABC的形状,并给出证明. C(-2,5) y 证明 : ?AB ? (2 ?1,3 ? 2) ? (1,1) AC ? (?2 ?1,5 ? 2) ? (?3,3) ?AB? AC ? 1? (?3) ? 1? 3 ? 0 B(2,3) A(1,2) 0 x ? AB ? AC ?三角形 ABC是直角三角形 . 向量数量积是否为零,是判断相应两条线段或直线的重 要方法之一 练习2:以原点和A(5,2) 为两个顶点作等腰直角三角形 OAB,?B=90?,求点B的坐标. 3 7 y 答案:B的坐标为( , ) B 2 2 7 3 或( , ? ) 2 2 O A x 四、逆向及综合运用 例3 (1)已知 a =(4,3),向量 b是 垂直于 a 的单位向量,求 b . (2)已知a ? 10, b ? (1,2),且a // b,求a的坐标. 3? (3)已知a ? (3,0), b ? (k ,5),且a与b的夹角为 , 4 求k的值. 3 4 3 4 答案:( 1 ) b ? ( ,? )或b ? (? , ). 5 5 5 5 (2)( 2, 2 2)或( ? 2, ? 2 2);(3)k ? ?5. 例4:已知 a =(1, 3),b =( 3+1, 3 ? 1), 则a与b的夹角是多少? 解:由a =(1, 3),b =( 3+1, 3 ? 1), 有 a b ? 1? ( 3 ? 1) ? 3 ? ( 3 ? 1) ? 4, a ? 2, b ? 2 2, 记a与b的夹角为θ,则 cos? ? 又∵0≤θ≤π,∴ cos? ? a b ab ? 4 评述:已知三角函数值求角时,应注意角的 范围的确定。 提高练习 1、已知OA ? (?3,1), OB ? (0,5),且 AC // OB, BC ? AB ,则点C的坐标为 29 C (?3, ) 3 2、已知A(1,2)、B(4、0)、C(8,6)、 D(5,8),则四边形ABCD的形状是矩

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