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luguofui:2019-2020年高中数学2.1向量的线性运算2.1.3向量的减法课后导练新人教B版必修

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2019-2020 年高中数学 2.1 向量的线性运算 2.1.3 向量的减法课后导练新

人教 B 版必修

基础达标

1.可以写成①;②;③;④.其中正确的是( )

A.①②

B.②③

C.③④

D.①④

解 析 : 由 三 角 形 法 则 知 ①④ 正 确 . 而

AO? OC ? AO? OC ? 2OC ? AC ? 2OC,OA? OC ? CA .

答案:D

2.化简下列各式,结果为零向量的个数是( )

① ②-+-

③④

A.1

B.2

C.3

D.4

解析:①==0.

②-+-=(+)-(+)=-=0.

③==0.

④ ? NP ? PM ? MN ? NM ? MN =0.

答案:D

3.若||=8,||=5,则||的取值范围是( )

A.[3,8]

B.(3,8)

C.[3,13]

解析:∵=-,当、同向时,||=8-5=3;

当,反向时,||=8+5=13;

当,不共线或有零向量时 3<||<13.

综上,知 3≤||≤13.

答案:C

4.△ABC 中,D、E、F 分别为 AB、BC、CA 的中点,等于(

A.

B.

C.

D.

D.(3,13) )

解析: AF ? DB ? AF ? AD ? DF ? BE .

答案:D

5.下列四式中,不能化简为的是( )

A.()+

B.(+)+(+)

C.+-

D.

解析:(+)+(+)=++=+.

答案:B

6.已知=a,=b,若||=12,||=5,且∠AOB=90°,则|a-b|=__________.

解析:|a-b|=|-|=||=.

答案:13

7. 在 边 长 为 1 的 正 方 形 ABCD 中 , 设 =a,=b,=c, 则

|a+b+c|=________,|a+c-b|=________,|c-a-b|=______.

解析:|a+b+c|=2|c|=,|a+c-b|=|(c-b)+a|=2|a|=2,|c-a-b|=0.

答案: 2 0

8.在平行四边形 ABCD 中,若|+|=|-|,则 ABCD 是__________(填正方形或矩形或菱形).

解析:由|+|=|-|,

即||=||,可得 ABCD 的对角线相等且为平行四边形,

因此可得 ABCD 为矩形.

答案:矩形

综合运用

9.(xx 全国高考,文 5) 已知向量 a,b 满足:|a|=1,|b|=2,|a-b|=2,则|a+b|等于( )

A.1

B.

C.

D.

解析:由|a+b|2+|a-b|2=2(|a|2+|b|2),

知|a+b|2+4=2(1+4),

故|a+b|=.

答案:D 10.平面上有三点 A,B,C,设 m=+,n=-,若 m,n 的长度恰好相等,则有( ) A.A,B,C 三点必在同一直线上 B.△ABC 必为等腰三角形且∠B 为顶角 C.△ABC 必为直角三角形且∠B=90° D.△ABC 必为等腰直角三角形
解析:如图,作 ABCD, 则+=, -=-=, ∵|m|=|n|, ∴||=||. ∴ABCD 为矩形. ∴△ABC 为直角三角形,∠B=90°. 答案:C 11.已知等腰直角△ABC,∠C=90°,M 为斜边的中点,设=a,=b,试用向量 a,b 表示、,,.
解:=-=a-b, ==a-b,

=+=++=b+a-b+a-b=2a-b, ==-(+)=-2(a-b)=2(b-a). 拓展探究 12.一艘船以 km/h 的速度向垂直于岸的方向行驶,而船的实际速度是 10 km/h,求水流的速度 和船行驶的方向(用与水流方向间的夹角表示).
解:如图所示,设表示船垂直于对岸行驶的速度,表示水流的速度,以,为边作 ABCD,则表示的 就是船实际航行的速度. 在 Rt△ABC 中, ||=10 km/h, ||=||= km/h,
∴||= | AC |2 ? | BC |2 ? 100 ? 75 (km/h).
∵tan∠CAB=,∴∠CAB=60°. 答:水流速度为 5 km/h,船行驶方向与水流方向夹角为 60°.


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