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www. gyao511.com:2019-2020学年高中数学 课时作业18 平面向量数量积的坐标表示 北师大版必修4

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2019-2020 学年高中数学 课时作业 18 平面向量数量积的坐标表示 北师大版必修 4

|基础巩固|(25 分钟,60 分)

一、选择题(每小题 5 分,共 25 分)

1.已知向量 a=(1,m),b=(3,-2),且(a+b)⊥b,则 m=( )

A.-8 B.-6

C.6

D.8

解析:由题可得 a+b=(4,m-2),又(a+b)⊥b,∴4×3-2×(m-2)=0,∴m=8.故

选 D.

答案:D

2.已知向量 a=(1, 3),b=(3,m),若向量 a,b 的夹角为π6 ,则实数 m 的值为(

)

A.2 3 B.- 3

C.0 D. 3

解析:由题意得|a|=2,|b|= 9+m2,a·b=3+ 3m=2 9+m2cosπ6 ,解得 m= 3, 选 D.
答案:D 3.若 a=(2,1),b=(3,4),则向量 a 在向量 b 方向上的射影的数量为( )

A.2 5 B.2

C. 5 D.10 解析:设 a,b 的夹角为 θ ,则|a|cosθ =|a|·|aa·||bb|=a|·b|b=2×3+5 1×4=2.

答案:B





→→

4.已知 O 为坐标原点,向量OA=(2,2),OB=(4,1),在 x 轴上有一点 P 使得AP·BP有

最小值,则点 P 的坐标是( )

A.(-3,0) B.(2,0)

C.(3,0) D.(4,0)





解析:设点 P 的坐标为(x,0),则AP=(x-2,-2),BP=(x-4,-1),

→→ AP·BP=(x-2)(x-4)+(-2)×(-1)

=x2-6x+10=(x-3)2+1,

→→ ∴当 x=3 时,AP·BP有最小值 1,

∴点 P 的坐标为(3,0).

答案:C

5.已知向量 a=(1,2),b=(2,-3).若向量 c 满足(c+a)∥b,c⊥(a+b),则 c=( )

A.???79,73??? B.???-73,-79???

C.???73,79??? D.???-79,-73???

解 析 :设 c = (x, y),则 c +a = (1+ x,2 + y), a + b = (3, - 1),由已知可得

?? +y + x+ =0 ???3x-y=0,

??x=-79 解得???y=-73,

即 c=???-79,-73???.

答案:D 二、填空题(每小题 5 分,共 15 分) 6.设 a=(m+1,-3),b=(1,m-1),若(a+b)⊥(a-b),则 m=________. 解析:a+b=(m+1,-3)+(1,m-1)=(m+2,m-4), a-b=(m+1,-3)-(1,m-1)=(m,-2-m), 因为(a+b)⊥(a-b),所以(a+b)·(a-b)=0,即(m+2,m-4)·(m,-m-2)=0, 所以 m2+2m-m2+2m+8=0,解得 m=-2. 答案:-2 7.已知平面向量 a=(1,2),b=(4,2),c=ma+b(m∈R),且 c 与 a 的夹角等于 c 与 b 的夹角,则 m=________.
解析:c=(m+4,2m+2),|a|= 5,|b|=2 5, 设 c,a 的夹角为 α ,c,b 的夹角为 θ , 又因为 cosα =|cc·||aa|,cosθ =|cc·||bb|, 由题意知c|·a|a=c|·b|b,即5m+5 8=8m2+520.
解得 m=2. 答案:2 8.

→→ 如图,在矩形 ABCD 中,AB= 2,BC=2,点 E 为 BC 的中点,点 F 在边 CD 上,若AB·AF

→→ = 2,则AE·BF的值是________.

解析:以 A 为原点,AB 所在直线为 x 轴,AD 所在直线为 y 轴建立平面直角坐标系,设

F(x,2),







所以AE=( 2,1),AF=(x,2),AB=( 2,0),

→→ 所以AB·AF= 2x= 2,

所以 x=1,所以 F(1,2),

→ 所以BF=(1,2)-( 2,0)=(1- 2,2),

→→ 所以AE·BF= 2.

答案: 2

三、解答题(每小题 10 分,共 20 分)

9.已知平面向量 a=(1,x),b=(2x+3,-x),x∈R.

(1)若 a⊥b,求 x 的值;

(2)若 a∥b,求|a-b|. 解析:(1)若 a⊥b,则 a·b=(1,x)·(2x+3,-x)=1×(2x+3)+x(-x)=0,即 x2 -2x-3=0,解得 x=-1 或 x=3. (2)若 a∥b,则 1×(-x)-x(2x+3)=0, 即 x(2x+4)=0,解得 x=0 或 x=-2. 当 x=0 时,a=(1,0),b=(3,0), |a-b|=|(1,0)-(3,0)|=|(-2,0)|=2. 当 x=-2 时,a=(1,-2),b=(-1,2),

|a-b|=|(1,-2)-(-1,2)|=|(2,-4)|=2 5.

10.已知向量 a=(1, 3),b=(-2,0). (1)求 a-b 的坐标以及 a-b 与 a 之间的夹角; (2)当 t∈[-1,1]时,求|a-tb|的取值范围.

解析:(1)因为向量 a=(1, 3),b=(-2,0),所以 a-b=(1, 3)-(-2,0)=(3,

3),

所以

cos〈a-b,a〉=|aa--bb|·|aa|=4

6

= 3

23.

因为〈a-b,a〉∈[0,π ],所以向量 a-b 与 a 的夹角为π6 .

(2)|a-tb|2=a2-2ta·b+t2b2=4t2+4t+4=4???t+12???2+3.易知当 t∈[-1,1]时,|a -tb|2∈[3,12],所以|a-tb|的取值范围是[ 3,2 3].

|能力提升|(20 分钟,40 分)

11.设点 A(4,2),B(a,8),C(2,a),O 为坐标原点.若四边形 OABC 是平行四边形,则

→→ 向量OA与OC之间的夹角为( )

A.π3 B.π4

C.π6 D.π2

→→ 解析:∵四边形 OABC 是平行四边形,∴OA=CB即(4-0,2-0)=(a-2,8-a),∴a=6.





又∵OA=(4,2),OC=(2,6),

→→

→→ ∴cos〈OA,OC〉=

OA·OC →→



4×2+2×6

2

42+22· 22+62= 2 ,

|OA|·|OC|

→→ 又〈OA,OC〉∈[0,π

],∴O→A与O→C的夹角为π4

.

答案:B 12.已知 a=(4,-3),b=(2,1),若 a+tb 与 b 的夹角为 45°,则实数 t=________. 解析:因为 a=(4,-3),b=(2,1), 所以 a+tb=(2t+4,t-3), 所以(a+tb)·b=5t+5.

又|a+tb|= t+ 2+ t- 2

= 5t2+10t+25,

|b|= 5,(a+tb)·b=|a+tb||b|cos45°,

所以 5t+5=

5t2+10t+25×

2 5× 2 ,

整理得 t2+2t-3=0, 解得 t=1 或 t=-3, 经检验知 t=-3 不成立,故 t=1.

答案:1 13.已知 a,b,c 是同一平面内的三个向量,其中 a=(1,2).

(1)若|c|=2 5,且 c∥a,求 c 的坐标;

(2)若|b|= 25,且 a+2b 与 2a-b 垂直,求 a 与 b 的夹角 θ .

解析:(1)由 a=(1,2),得|a|= 12+22= 5,

又|c|=2 5,所以|c|=2|a|. 又因为 c∥a,所以 c=±2a, 所以 c=(2,4)或 c=(-2,-4). (2)因为 a+2b 与 2a-b 垂直, 所以(a+2b)·(2a-b)=0,

即 2|a|2+3a·b-2|b|2=0,将|a|= 5,|b|= 25代入,得 a·b=-52.

a·b 所以 cosθ =|a|·|b|=-1,

又由 θ ∈[0,π ],得 θ =π , 即 a 与 b 的夹角为 π . 14.已知三点 A(2,1),B(3,2),D(-1,4). (1)求证:AB⊥AD; (2)要使四边形 ABCD 为矩形,求点 C 的坐标并求矩形 ABCD 的对角线的长度.





解析:(1)∵A(2,1),B(3,2),D(-1,4),∴AB=(1,1),AD=(-3,3).

→→ 则AB·AD=1×(-3)+1×3=0,

→→ ∴AB⊥AD,即 AB⊥AD.

→→

→→

(2)∵AB⊥AD,四边形 ABCD 为矩形,∴AB=DC.

设 C 点的坐标为(x,y),则→DC=(x+1,y-4),从而有?????xy+ -14= =11, 即?????xy= =05, ∴C





点的坐标为(0,5).AC=(-2,4),|AC|= - 2+42=2 5,

∴矩形 ABCD 的对角线的长度为 2 5.


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以上会员名单排名不分前后