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2019-2020 年高一数学平面向量的数量积的坐标表示教案
教材:平面向量的数量积的坐标表示
目的:要求学生掌握平面向量数量积的坐标表示,掌握向量垂直的坐标表示的充要条件。 过程:
一、 复习: 1.平面向量的坐标表示及加、减、实数与向量的乘积的坐标表示 2.平面向量数量积的运算 3.两平面向量垂直的充要条件 4.两向量共线的坐标表示:
二、 课题:平面两向量数量 积的坐标表示 1. 设 a = (x1, y1),b = (x2, y2),x 轴上单位向量 i,y 轴上单位向量 j, 则:i?i = 1,j?j = 1,i?j = j?i = 0 2. 推导坐标公式: ∵a = x1i + y1j, b = x2i + y2j ∴a?b = (x1i + y1j )(x2i + y2j) = x1x2i2 + x1y1i?j + x2y1i?j + y1y2j2 = x1x2 + y1y2 从而 获得公式:a?b = x1x2 + y1y2 例一、 设 a = (5, ?7),b = (?6, ?4),求 a?b 解:a?b = 5×(?6) + (?7)×(?4) = ?30 + 28 = ?2 3. 长度、角度、垂直的坐标表示
1?a = (x, y) ? |a|2 = x2 + y2 ? |a| = x2 ? y 2

2?若 A = (x1, y1),B = (x2, y2),则 AB = (x1 ? x2 )2 ? ( y1 ? y2 )2

3? cos? = a ? b ?

x1 x2 ? y1 y2

| a |?| b |

x12 ? y12 x2 2 ? y2 2

4?∵a?b ? a?b = 0 即 x1x2 + y y1 2 = 0(注意与向量共线的坐标表示原则) 4. 例二、已知 A(1, 2),B(2, 3),C(?2, 5), 求证:△ABC 是直角三角形。

证:∵ AB =(2?1, 3?2) = (1, 1), AC = (?2?1, 5?2) = (?3, 3)

∴ AB ? AC =1×(?3) + 1×3 = 0 ∴ AB ? AC

∴△ABC 是直角三角形

三、补充例题:处理《教学与测试》P153 第 73 课

例三、已知 a = (3, ?1),b = (1, 2),求满足 x?a = 9 与 x?b = ?4 的向量 x。 解:设 x = (t, s),

由 x?a = 9 ? 3t ? s = 9

t=2

由 x?a = 9 ? 3t ? s = 9 ? s = ?3

∴x = (2, ?3)

例四、如图,以原点和 A(5, 2)为顶点作等腰直角△OAB,使?B = 90?,

求点 B 和向量 AB 的坐标。

B A

解:设 B 点坐标(x, y),则 OB = (x, y), AB = (x?5, y?2) O

∵ OB ? AB ∴x(x?5) + y(y?2) = 0 即:x2 + y2 ?5x ? 2y = 0

又∵| OB | = | AB | ∴x2 + y2 = (x?5)2 + (y?2)2 即:10x + 4y = 29



?x2 ?

?

y2

?

5x

?

?10x ? 4 y ? 29

2y

?

0

?

? ??x1 ?

? ??

y1

? ?

7 2 ?

3 2

? 或 ??
? ?

x2 y2

? ?

3
2 7
2

∴B 点坐标 (7 ,? 3) 或 ( 3 , 7) ; AB = (? 3 ,? 7) 或 (? 7 , 3)

2 2 22

22

22

例五、在△ABC 中, AB =(2, 3), AC =(1, k),且△ABC 的一个 内角为直角, 求 k 值。
解:当 A = 90?时, AB ? AC = 0,∴2×1 +3×k = 0 ∴k = ? 3 2

当 B = 90?时, AB ? BC = 0, BC = AC ? AB = (1?2, k?3) = (?1, k?3) ∴2×(?1) +3×(k?3) = 0 ∴k = 11 3

当 C = 90?时, AC ? BC = 0,∴?1 + k(k?3) = 0
四、小结:两向量数量积的坐标表示 长度、夹角、垂直的坐标表示
五、 作业: P121 练习及习题 5.7 《教学与测试》P154 5、6、7、8,思考题

∴k = 3 ? 13 2


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