36678com:高一数学课件:平面向量数量积的坐标表示_图文

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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

?j j = | j |2 = 1

? ? i j = j i = 0
∵a = x1 i + y1 j ,b = x2 i + y2 j

B(x2,y2)
b

A(x1,y1)

∴a?b =( x1 i + y1 j)?(x2 i + y2 j)

ja

= x1 x2 i2 + x1 y2 i j + y1 x2 j i + y1 y2 j2

Oi

X

= x1 x2 + y1 y2

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 | = x2? y2 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)

AC = (-2 -1,5 -2)= (-3,3)C

Y

∴AB ? AC = 1╳(-3)+ 1╳ 3 = 0

∴AB⊥AC

B

∴Δ ABC是直角三角形

A

O
注:两个向量的数量积是否为零是判断相应的

X

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。

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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