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课下能力提升(二十一)

平面向量数量积的坐标表示

一、选择题

1.若向量 a=(1,2),b=(1,-1),则 2a+b 与 a-b 的夹角等于( )

A.-π4

B.π6

C.π4

D.3π4

2.已知向量 a=(3,4),b=(2,-1),如果向量 a+xb 与-b 垂直,则 x 的值为( )

A.-25 B.233

C.233

D.2

3.已知向量 a=(2,1),a·b=10,|a+b|=5 2,则|b|=( ) A. 5 B. 10 C.5 D.25

4.已知 AB =(4,2), AC =(k,-2),若△ABC 为直角三角形,则 k 等于( )
A.1 B.6
C.1 或 6 D.1 或 2 或 6 二、填空题 5.(安徽高考)设向量 a=(1,2m),b=(m+1,1),c=(2,m).若(a+c)⊥b,则|a|=________. 6.(新课标全国卷Ⅰ)已知两个单位向量 a,b 的夹角为 60°,c=ta+(1-t)b.若 b·c=0, 则 t=________. 7.已知向量 a=(1,2),b=(2,-3).若向量 c 满足(c+a)∥b,c⊥(a+b),则 c=________. 8.已知 a=(1,3),b=(1,1),c=a+λ b,若 a 和 c 的夹角是锐角,则 λ 的取值范围是 ________. 三、解答题 9.已知向量 a 是以点 A(3,-1)为始点,且与向量 b=(-3,4)垂直的单位向量,求 a 的终 点坐标.

-1-

10.已知△ABC 中,A(2,4),B(-1,-2),C(4,3),BC 边上的高为 AD. (1)求证:AB⊥AC; (2)求点 D 和向量 AD 的坐标; (3)设∠ABC=θ ,求 cos θ .
答案 1.解析:选 C 因为 2a+b=(2,4)+(1,-1)=(3,3), a-b=(0,3), 所以|2a+b|=3 2,|a-b|=3. 设 2a+b 与 a-b 的夹角为 θ , 则 cos θ =(2a|2+ab+)b· ||( a-ab-|b)=(3,33)·2×(30,3)= 22, 又 θ ∈[0,π ], 所以 θ =π4 . 2.解析:选 A ∵a+xb=(3,4)+x(2,-1)=(3+2x,4-x), -b=(-2,1),且(a+xb)⊥(-b), ∴-2(3+2x)+(4-x)=0,得 x=-25. 3.解析:选 C 法一:设 b=(x,y), 则 a·b=2x+y=10 ①, 又 a+b=(x+2,y+1),|a+b|=5 2, ∴(x+2)2+(y+1)2=50 ② ①与②联立得?????xy= =34, ,或?????xy= =50, . ∴|b|= x2+y2=5.
-2-

法二:由|a+b|=5 2得 a2+2a·b+b2=50, 即 5+20+b2=50 ∴b2=25|b|=5.
4.解析:选 C 当 A=90°时, AC ⊥ AB ,则 4k-4=0,k=1;
当 B=90°时, AB ⊥ BC ,又 BC = AC - AB =(k-4,-4)
∴4(k-4)+2×(-4)=0 解得 k=6;
当 C=90°时, AC ⊥ BC ,则 k(k-4)+(-2)×(-4)=0
即 k2-4k+8=0,无解. 故 k=1 或 6. 5.解析:由题意知,a+c=(3,3m), (a+c)·b=3(m+1)+3m=0,解得 m=-12, 即 a=(1,-1),|a|= 12+(-1)2= 2. 答案: 2 6.解析:本题考查平面向量的数量积运算,意在考查考生的运算求解能力.根据数量积 b·c =0,把已知两向量的夹角转化到两向量数量积的运算中.因为向量 a,b 为单位向量,所以 b2= 1,又向量 a,b 的夹角为 60°,所以 a·b=12,由 b·c=0 得 b·[ta+(1-t)b]=0,即 ta·b +(1-t)b2=0,所以12t+(1-t)=0,所以 t=2. 答案:2 7.解析:本题主要考查向量的基本知识及运算.由题意,将 b·c=[ta+(1-t)b]·b 整理, 得 ta·b+(1-t)=0,又 a·b=12,所以 t=2. 答案:2 7.解析:设 c=(x,y),则 c+a=(x+1,y+2). 又(c+a)∥b, ∴2(y+2)+3(x+1)=0.① 又 c⊥(a+b), ∴(x,y)·(3,-1)=3x-y=0.② 解①②得 x=-79,y=-73.
-3-

答案:???-79,-73???

8.解析:由条件得,c=(1+λ ,3+λ ),从而

??a×c=1+λ +3(3+λ )>0,

?1+λ ?? 1

≠3+3λ



? λ ∈???-52,0???∪(0,+∞).

答案:???-52,0???∪(0,+∞)

9.解:∵b 是直线 y=-43x 的方向向量,且 a⊥b.

∴a 是直线 y=34x 的方向向量.

∴可设 a=λ (1,34)=(λ ,34λ ).

由|a|=1,

得 λ 2+196λ 2=1. 解得 λ =±45, ∴a=(45,35)或 a=(-45,-35).

设 a 的终点坐标为(x,y)

??x-3=54, ??x-3=-45, 则???y+1=35,或???y+1=-35.
??x=159, ??x=151, 即???y=-25,或???y=-85.
∴a 的终点坐标是(159,-25)或(151,-85).

10.

-4-

∴5(x+1)=5(y+2),②

由①②解得 x=72,y=52,



D

75 点坐标为(2,2),

-5-


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