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5.7 平面向量数量积的坐标表示 教学目标: 1、平面向量数量积的坐标表示 2、两个向量垂直的坐标表示的充要条件 3、平面内两点间的距离公式 4、运用两个向量的数量积的坐标表示解决 处理有关长度垂直的几个问题 5、两个向量垂直与平行的充要条件的区别 前提测评: 1、已知A(3,5),B(6,9),则 AB = (3,4) 2、已知AB = a,a = (1,3),A = (1,5),则B 点的坐标为 (0,8) 3、(1)、已知a =(2,4),b =(5,2),则 a + b = (3,6) ,a -b = (-7,2) (2)、已知| a |= 8,| b |= 6,a和 b的夹角为600, 则a ? b = 24 , a ? a = 64 ,| a |= 8 (3)、a⊥ b则a ? b = 0 在直角坐标系中,已知两个非零向量a = (x1,y1), b = (x2,y2), 如何用a 与b的坐标表示a ? b 先看x轴上的单位向量i, y轴上的单位向量j i i = | i |2 = 1 Y ? i? j = j ? i = 0 ? j j = | j |2 = 1 B(x2,y2) ∵a = x1 i + y1 j ,b = x2 i + y2 j b j O A(x1,y1) ? x2 i + y2 j) ∴a ? b =( x1 i + y1 j)( = x1 x2 i2 + x1 y2 i j + y1 x2 j i + y1 y2 j2 = x1 x2 + y1 y2 a X i 两个向量的数量积等于 它们的对应坐标乘积的和 即: a ? b = x1 x2 + y1 y2 例1、设a = (5,7),b = (6,4),求a ? b 特殊地:1、设a = (x,y),则a ? a = | a |2 = x 2+ y2 或: | a | = 2? y2 x 2、设a = AB,若A(x1,y1),B(x2,y2), 则 | a | =| AB|= ( x1? x2) 2 ? ( y1? y2)2 即是平面内两点间的距离公式 3、设a = (x1,y1),b = (x2,y2),则 a ⊥b <===> x1 x2 + y1 y2 = 0 a ∥b <===>x1 y2 -x 2y1=0 例2、已知A(1、2),B(2,3),C(?2,5),求 证Δ ABC是直角三角形 证明: ∵AB = (2 ? 1,3 ? 2)= (1,1) Y AC = (?2 ? 1,5 ? 2)= (?3,3) C ∴AB ? AC = 1╳(?3)+ 1╳ 3 = 0 ∴AB⊥AC ∴Δ ABC是直角三角形 O A B 注:两个向量的数量积是否为零是判断相应的 两条直线是否垂直的重要方法之一。 如证明四边形是矩形,三角形的高,菱形对角线 垂直等。 X 小结: 1、数量积的坐标表示 2、垂直的充要条件 3、平面内的两点间距离 达标测评:1、已知a = (?3,4),b = (5,2),求a? b ,| a |,| b |。 2、a = (2,3),b = (?2,4),C = (?1,?2)求a b , (a + b) (a - b),a (b + C),(a + b)2 3× 、已知a = (?2, ×4),b = (1,?2),则a 与b的关系是 A、不共线 B、垂直 C、共线同向 D、共线反向 4、以A(2,5),B(5,2),C(10,7)为顶点的三角 形的形状是 × √ A、等腰三角形 B、直角三角形 C、等腰直角三角形 D、等腰三角形或直角三角形 1、 a ? b= -7, | a |= 5, | b|= 29 , ? ? ? × √ 2、 a ? b = 8, (a + b)?(a - b)= - 7, a ?(b + C)= 0 (a + b)2 = 49 作业:P1215.7 1、2、3、4

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