1 . (1)已知
,则![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8690c5b4f4e2f9f13e2a94da1a688ecf.png)
______ ;
(2)若
,
,则![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9b059c195ed8806d208984e89f400692.png)
______ ;
(3)![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2bbf6266593c04e6cc80b462da609828.png)
______ .
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4a8c9f990fc696c4b27d02cb9e92fe9c.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8690c5b4f4e2f9f13e2a94da1a688ecf.png)
(2)若
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/bb39c8e18752c1348b067e83b3022e46.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8504e8973f9af7a594d199e1eca01ea4.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9b059c195ed8806d208984e89f400692.png)
(3)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2bbf6266593c04e6cc80b462da609828.png)
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23-24高一下·全国·课前预习
2 . 定义:已知两个非零向量
,
,O是平面上的任意一点,作
=
,
=
,则∠AOB=θ(0≤θ≤π)叫做向量
与
的夹角.
注意:①当θ=0时,向量
与![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b45ba716f03748c19b7ce2f99af536ab.png)
_____ ;
时,向量
与![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b45ba716f03748c19b7ce2f99af536ab.png)
_____ ,记作
⊥
;
③当θ=π时,向量
与![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b45ba716f03748c19b7ce2f99af536ab.png)
______ .
注意:只有两个向量的起点重合时所对应的角才是两向量的夹角,如图所示,∠BAC不是向量
与
的夹角.作
=
,则∠BAD才是向量
与
的夹角.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/64c5562bd4d1b54424330cb6329cd79d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b45ba716f03748c19b7ce2f99af536ab.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a4f605ec0729ce6d72237ad662a06862.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/64c5562bd4d1b54424330cb6329cd79d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6fc9656d8286c4d6fa309d6ae347c89e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b45ba716f03748c19b7ce2f99af536ab.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/64c5562bd4d1b54424330cb6329cd79d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b45ba716f03748c19b7ce2f99af536ab.png)
注意:①当θ=0时,向量
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/64c5562bd4d1b54424330cb6329cd79d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b45ba716f03748c19b7ce2f99af536ab.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d49f8a63ddbca52039fa9ab44cda6b29.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/64c5562bd4d1b54424330cb6329cd79d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b45ba716f03748c19b7ce2f99af536ab.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/64c5562bd4d1b54424330cb6329cd79d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b45ba716f03748c19b7ce2f99af536ab.png)
③当θ=π时,向量
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/64c5562bd4d1b54424330cb6329cd79d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b45ba716f03748c19b7ce2f99af536ab.png)
注意:只有两个向量的起点重合时所对应的角才是两向量的夹角,如图所示,∠BAC不是向量
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/cbaeae7045ad94158cdf5ae97073bc17.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/abcb5d89b04570ceda2c29e11cb27a57.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/304a7f07db2ec637baadf8f0ab91c85c.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/cbaeae7045ad94158cdf5ae97073bc17.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/cbaeae7045ad94158cdf5ae97073bc17.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/abcb5d89b04570ceda2c29e11cb27a57.png)
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23-24高一下·全国·课后作业
3 . 用简单随机抽样的方法从含有6个个体的总体中,抽取一个样本量为2的样本,某一个个体a“第一次被抽到的概率”、“第二次被抽到的概率”、“在整个抽样过程中被抽到的概率”分别是________ 、________ 、________ .
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23-24高一下·全国·课前预习
4 . 向量的概念和表示方法
(1)向量:在数学中,我们把既有____ 又有_____ 的量叫做向量.
(2)向量的表示
①表示工具——有向线段.
有向线段包含三个要素:______ ,______ ,______ .
②表示方法:
向量可以用__________ 表示,向量
的大小称为向量
的____ (或称模),记作______ .向量可以用字母
…表示,也可以用有向线段的起点和终点字母表示,如:
,
.
(1)向量:在数学中,我们把既有
(2)向量的表示
①表示工具——有向线段.
有向线段包含三个要素:
②表示方法:
向量可以用
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/abcb5d89b04570ceda2c29e11cb27a57.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/abcb5d89b04570ceda2c29e11cb27a57.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2e12e95f703ad30ab9a3d38376830989.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/abcb5d89b04570ceda2c29e11cb27a57.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/bc1070a28cb9cb8553c29747d1993b16.png)
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5 . 向量的模及两个特殊向量
(1)向量的模(长度):向量
的大小,称为向量
的______ (或称模),记作______ .
(2)零向量:长度为______ 的向量,记作
.
(3)单位向量:长度等于__________________ 的向量.
(1)向量的模(长度):向量
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d1fe2d802f2b37e7db198c5a3c1df9a4.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d1fe2d802f2b37e7db198c5a3c1df9a4.png)
(2)零向量:长度为
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a85b26387e5425ca06cf14fcc15dc1b1.png)
(3)单位向量:长度等于
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23-24高一下·全国·课前预习
6 . 向量加法的定义:
求两个向量和的运算,叫做向量的加法.两个向量的和仍然是一个________ .
对于零向量与任意向量
,规定:![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1891c27d3ee38a035f53e7289efff263.png)
________ =________ .
求两个向量和的运算,叫做向量的加法.两个向量的和仍然是一个
对于零向量与任意向量
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7a2f4b1178f68bd147d1a2a6acd04435.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1891c27d3ee38a035f53e7289efff263.png)
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7 . 时间经过
(时),时针转了______ 度,等于______ 弧度;若时针长度是1厘米,则时针
(时)转出的扇形面积是______ 平方厘米.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5c63f6b1652d3e830ed2dbc65253d043.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/15694733427d51a02995963186c5c427.png)
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2024高一下·全国·专题练习
8 . 一个建筑物上部为四棱锥,下部为长方体,且四棱锥的底面与长方体的上底面重合,已知长方体的长、宽、高分别为20
,5
,10
,四棱锥的高为8
,若利用斜二测画法按1∶500的比例画出它的直观图,那么直观图中长方体的长为________
,宽为________
,建筑物的高为________
.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e15e00f40396e914d1d9955bd7785f1f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e15e00f40396e914d1d9955bd7785f1f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e15e00f40396e914d1d9955bd7785f1f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e15e00f40396e914d1d9955bd7785f1f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9efa9fbcfb9595e2f031aa691db4564b.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9efa9fbcfb9595e2f031aa691db4564b.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9efa9fbcfb9595e2f031aa691db4564b.png)
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2024高三·全国·专题练习
9 . (1)已知{an},{bn}都是等差数列.若a1+b10=9,a3+b8=15,则a5+b6=______ .
(2)设等差数列{an}的前n项和为Sn.若S3=9,S6=36,则a7+a8+a9=______ .
(3)已知Sn是等差数列{an}的前n项和.若a1=-100,
-
=6,则S100=______ .
(2)设等差数列{an}的前n项和为Sn.若S3=9,S6=36,则a7+a8+a9=
(3)已知Sn是等差数列{an}的前n项和.若a1=-100,
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a60b42c8031db209eb0605f1efe2fbdf.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e5e3f5a061e2cf401b9368b15882abe3.png)
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24-25高一上·全国·课后作业
解题方法
10 . 填空:(1)被9除余2的所有整数组成的集合可表示为______ ;
(2)不等式组
的解集为A,则![](https://staticzujuan.xkw.com/quesimg/Upload/formula/efc2ddf695c270d11bda83afcf9982e9.png)
______ ;
(3)已知集合
,
,则![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a95ecdb34ced00248d02a700d3c8af07.png)
______ ;
(4)满足
的集合B的个数是______ ;
(5)已知集合
或
,
,则
与
的关系是______ .
(2)不等式组
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/42685244fca608665a87e2f10b8a322e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/efc2ddf695c270d11bda83afcf9982e9.png)
(3)已知集合
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ef257e2a11673facf09edaa806da69d2.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ba26e7cc7485c6d62ab40aa704485387.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a95ecdb34ced00248d02a700d3c8af07.png)
(4)满足
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/40c682336d9b7ef46430c4970f52b27e.png)
(5)已知集合
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/73ca8236c89d9cd1aeda40a6e6ed70ca.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d49f1b5bc97211d6feaa1b6538ec50c0.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/dfe672a0a9589b8290ac05cfaaac8e72.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/afdff52c0a2b46e0077fa82a17806bae.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d4150f7b0bd8bc03f856c4ac563507ce.png)
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