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870826com:高中数学 第二章平面向量的数量积 2.3.2 向量数量积的运算律练习 新人教B版必修4

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2.3.2 向量数量积的运算律

课时过关·能力提升 1.已知两个非零向量 a,b 满足|a+b|=|a-b|,则下面结论正确的是( )

A.a∥b

B.a⊥b

C.|a|=|b|

D.a+b=a-b

解析:|a+b|2=|a|2+2a·b+|b|2,

|a-b|2=|a|2-2a·b+|b|2.

因为|a+b|=|a-b|,

所以|a|2+2a·b+|b|2=|a|2-2a·b+|b|2,

即 2a·b=-2a·b,

所以 a·b=0,

所以 a⊥b.故选 B.

答案:B

2.设向量 a,b,c 满足 a+b+c=0,a⊥b,|a|=1,|b|=2,则|c|2 等于( )

A.1

B.2

C.4

D.5

解析:由 a+b+c=0 得 c=-(a+b),

于是|c|2=|-(a+b)|2=|a|2+2a·b+|b|2=1+4=5.

答案:D

3.已知|a|=3,|b|=4,且(a+kb)⊥(a-kb),则实数 k 的值为( )

A.±

B.±

C.±

D.±

解析:由(a+kb)⊥(a-kb)知(a+kb)·(a-kb)=0,

即|a|2-k2|b|2=0,

因此 9-16k2=0,所以 k=± . 答案:A

尚水作品

4.已知 a,b 是非零向量,满足(a-2b)⊥a,(b-2a)⊥b,则 a 与 b 的夹角是( )

A.

B.

C.

D.

解析:由已知得(a-2b)·a=0, 因此|a|2-2a·b=0. 同理(b-2a)·b=0,即|b|2-2a·b=0,

于是有|a|=|b|,且 a·b=

|a|2,

从而 cos<a,b>=

,

又<a,b>∈[0,π ],所以 a 与 b 的夹角为 . 答案:B

5.如图,在菱形 ABCD 中,下列关系式不正确的是( ) A.

B.(

)⊥(

)

C.(

)·(

)=0

D.

解析:由于

,

所以 答案:D 6.

,故 D 项不正确.

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如图,在△ABC 中,AD⊥AB,

,| |=1,则

等于 ( )

A.2

B.

C.

D.

解析:由图可得

=(



.

∵AD⊥AB,∴

=0.

又∵

,





=0+

|2=

.∴

=0+

.

答案:D

7.已知平面向量 a,b 满足|a|=1,|b|=2,a 与 b 的夹角为 ,以 a,b 为邻边作平行四边形,则此平行四

边形的两条对角线中较短的一条对角线的长度为

.

答案:

8.已知 a+b=2i-8j,a-b=-8i+16j,其中 i·j=0,|i|=|j|=1,则 a·b=

.

答案:-63

9.设 O,A,B,C 为平面上的四个点, =a, =b, =c,且 a+b+c=0,a·b=b·c=c·a=-1,则

|a|+|b|+|c|=

.

答案:3

10.在边长为 1 的等边三角形 ABC 中,设 =2

=3 ,则

=

.

解析:由已知得

),

,

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所以



-|

|2-

=- . 答案:★11.设 a⊥b,且|a|=2,|b|=1,k,t 是两个不同时为零的实数. (1)若 x=a+(t-3)b 与 y=-ka+tb 垂直,求 k 关于 t 的函数关系式 k=f(t);
(2)求出函数 k=f(t)的最小值. 解:(1)∵a⊥b,∴a·b=0.
又 x⊥y,∴x·y=0, 即[a+(t-3)b]·(-ka+tb)=0, ∴-ka2-k(t-3)a·b+ta·b+t(t-3)b2=0. ∵|a|=2,|b|=1, ∴-4k+t2-3t=0,
∴k= (t2-3t)(t≠0),

即 k=f(t)= (t2-3t)(t≠0).

(2)由(1)知 k=f(t)= (t2-3t)

=

,

故函数 k=f(t)的最小值为-

.

★12.已知|a|= ,|b|=1,向量 a 与 b 的夹角为 45°,求使向量(2a+λ b)与(λ a-3b)的夹角为锐角 的 λ 的取值范围. 解:设向量(2a+λ b)与(λ a-3b)的夹角为 θ .

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∵两向量的夹角为锐角,



>0,

∴(2a+λ b)·(λ a-3b)>0, 即 2λ a2+(λ 2-6)a·b-3λ b2>0. ∵a2=|a|2=2,b2=|b|2=1,

a·b=|a||b|cos 45°=

×1×

=1,

∴4λ +λ 2-6-3λ >0, 即 λ 2+λ -6>0,∴λ <-3 或 λ >2. 设 2a+λ b=k(λ a-3b)=kλ a-3kb,



∴λ 2=-6,则 λ 不存在,

即向量(2a+λ b)与(λ a-3b)不共线. ∴使向量(2a+λ b)与(λ a-3b)的夹角为锐角的 λ 的取值范围为 λ <-3 或 λ >2.

尚水作品


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