108hh新地址:高中数学第二章平面向量2.4.2平面向量数量积的坐标表示、模、夹角教案新人教A版必修4

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2.4.2 平面向量数量积的坐标表示、模、夹角
1.知识与技能 (1)掌握平面向量数量积坐标运算规律; (2)能利用数量积的 5 个重要性质及数量积运算规律解决有关问题; (3)掌握两个向量共线、垂直的几何判断,会证明两向量垂直,以及能解决一些简单问题. 2.过程与方法 通过对平面向量数量积的坐标表示的探究学习,培养学生的独立解决问题的能力和体会化归与 转化的数学思想方法. 3.情感、态度与价值观 通过对本节内容的学习和运用实践,培养学生的探索精神和应用意识,体会数学的科学价值和 应用价值.
重点:平面向量数量积的坐标表示及其运算. 难点:运用向量坐标运算求解向量垂直、夹角等相关问题.
1.已知平面向量 a=(3,4),b=(9,x),c=(4,y),且 a∥b,a⊥c. (1)求 b 和 c; (2)若 m=2a-b,n=a+c,求向量 m,n 的夹角的大小. 解:(1)∵a∥b,∴3x-36=0.∴x=12. ∵a⊥c,∴3×4+4y=0.∴y=-3. ∴b=(9,12),c=(4,-3). (2)m=2a-b=(6,8)-(9,12)=(-3,-4), n=a+c=(3,4)+(4,-3)=(7,1), 设 m,n 的夹角为 θ ,
则 cos θ =

=

=-

.

∵θ ∈[0,π ],∴θ =

,即 m,n 的夹角为

.

1

2.已知三个点 A(2,1),B(3,2),D(-1,4). (1)求证:AB⊥AD; (2)若四边形 ABCD 为矩形,求点 C 的坐标及矩形 ABCD 两对角线所夹的锐角的余弦值. (1)证明:∵A(2,1),B(3,2),D(-1,4),



=(1,1),

=(-3,3).



=1×(-3)+1×3=0,



,∴AB⊥AD.

(2)解:∵

,四边形 ABCD 为矩形,



.

设点 C 的坐标为(x,y),则

=(x+1,y-4).



=(1,1),∴

解得 ∴点 C 的坐标为(0,5).



=(-2,4),

=(-4,2),

∴|

|=2

,|

|=2



的夹角为 θ ,

=8+8=16.

则 cos θ =

.

故矩形 ABCD 的两条对角线所夹的锐角的余弦值为

.

2


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