1 . 已知复数
满足
.
(1)求证:
;
(2)若
的虚部为正数,求
,
,
,
,根据
的规律,求出
的值(不需要证明).
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/81dea63b8ce3e51adf66cf7b9982a248.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8646eaa05bfde39d27813c301a076420.png)
(1)求证:
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c5c6b89b69721e00cdfb30bef61a4916.png)
(2)若
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/81dea63b8ce3e51adf66cf7b9982a248.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/81dea63b8ce3e51adf66cf7b9982a248.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9b0a89e3c30f6e4d4c5db4378b05d987.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d800f03de80068a1172beac3a2c75587.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3336c8ed5361c10c37300e41e03f9f2f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6ccdb989a252ffac6e46c5a80979043c.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0ace3db19b2dd8d613cfbbb1abdfddc3.png)
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2 . 现新定义两个复数
(
、
)和
(
、
)之间的一个新运算
,其运算法则为:
.
(1)请证明新运算
对于复数的加法满足分配律,即求证:
;
(2)设运算
为运算
的逆运算,请推导运算
的运算法则.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4734ba5631121fac6a8488d56ab05d58.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e72adb45c60c2f63b46e65ff787302bf.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ebfabd3af39419289bfeb15e2ffc672c.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d14d00cc1a2b2d219e9140c9c61ae01b.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3e88093a749c0d46e0ee931ecfaff925.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5a5103387b9d41429d9882ea1f2e9e3d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/36e16415b61722f9961e412386e6819f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3ab1f1bc414abe3f0c2ccbc537af2a5e.png)
(1)请证明新运算
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/36e16415b61722f9961e412386e6819f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c214bf6ceecb02886ca24322fe2ddb15.png)
(2)设运算
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3334218505dc667679059f10d5c5b93a.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/36e16415b61722f9961e412386e6819f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3334218505dc667679059f10d5c5b93a.png)
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2020-07-16更新
|
325次组卷
|
6卷引用:第三章 数系的扩充与复数的引入【专项训练】-2020-2021学年高二数学(理)下学期期末专项复习(人教A版选修2-2)
(已下线)第三章 数系的扩充与复数的引入【专项训练】-2020-2021学年高二数学(理)下学期期末专项复习(人教A版选修2-2)苏教版(2019) 必修第二册 必杀技 专练1 新定义、新情境专练(已下线)7.2.2 复数的乘、除运算 (精讲)(2)-【精讲精练】2022-2023学年高一数学下学期同步精讲精练(人教A版2019必修第二册)上海市静安区2019-2020学年高二下学期期末数学试题沪教版(2020) 必修第二册 同步跟踪练习 第9章 复数 9.1~9.2 阶段综合训练沪教版(2020) 必修第二册 同步跟踪练习 第9章 9.1~9.2阶段综合训练
名校
3 . 在复平面内复数
,
所对应的点为
,
,
为坐标原点,
是虚数单位.
(1)
,
,计算
与
;
(2)设
,![](https://staticzujuan.xkw.com/quesimg/Upload/formula/368f9e12546277731776041c73dbe58c.png)
,求证:
,并指出向量
,
满足什么条件时该不等式取等号.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/af68f652b4c13657ffddf3c9e7eb262b.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/fa224ed9be8766a4d0b5138bd57de0f0.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4bb8477bcc87b1401970171bf57b9ecb.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b2c1fd680a5d355178273c6d6025eb80.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1dde8112e8eb968fd042418dd632759e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3a7035cd4adda5d72a9fc9f9fda75995.png)
(1)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9c552c2a57bd5d204c2d700ef1112554.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1435039f4745c7e6fbc266636e414705.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6da4e6752d8c8a0705194f2b2f16ab5d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7f75805768bce2c1699aa5f9e33adbf4.png)
(2)设
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5a0bf9b2a7378e73e9fd06c693bfda07.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/368f9e12546277731776041c73dbe58c.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/32d3995815fb78d6abdacc08145a89b3.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b49f7ebf36aba9ca166881222ca6aa71.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a30deb1f343048675b9b231620369668.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a1640d3fff861f45c5eb4019943b000f.png)
您最近一年使用:0次
2024高一下·江苏·专题练习
4 . 设
,求证:
(1)
;
(2)
.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ed3a7e1ea06b2eb0061ad24605eb7fdd.png)
(1)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b3cd88b6123407e39b571d2daf0a5e55.png)
(2)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ba1a29a88a4ac9e429cac9600adaede1.png)
您最近一年使用:0次
解题方法
5 . 在复平面内复数
所对应的点为
,O为坐标原点,i是虚数单位.
(1)
,计算
与
;
(2)设
,求证:
,并指出向量
满足什么条件时该不等式取等号.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/90d860cb86e1467ac24010aecfc7a425.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/38033198bf936b904a8c74db67e4cdcf.png)
(1)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/35691b17b42b5fd4bfc4598240071cb9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6da4e6752d8c8a0705194f2b2f16ab5d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7f75805768bce2c1699aa5f9e33adbf4.png)
(2)设
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d9b3e4d91a97797c4c090960ad88bd62.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b49f7ebf36aba9ca166881222ca6aa71.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1a5cdedb6f4384fda29fb4508ba6fcc5.png)
您最近一年使用:0次
2024-03-19更新
|
359次组卷
|
21卷引用:热点04 平面向量、复数-2021年高考数学【热点·重点·难点】专练(上海专用)
(已下线)热点04 平面向量、复数-2021年高考数学【热点·重点·难点】专练(上海专用)沪教版(2020) 必修第二册 高效课堂 册末测试卷(已下线)专题14 复数(模拟练)(已下线)7.1.2 复数的几何意义(分层作业)-【上好课】2022-2023学年高一数学同步备课系列(人教A版2019必修第二册)(已下线)模块三 专题5 大题分类练(复数)基础夯实练(人教A)(已下线)模块三 专题6(复数)基础夯实练(北师大版)(已下线)模块三 专题7 大题分类练(复数)基础夯实练(苏教版)(已下线)第十二章 复数(单元重点综合测试)-单元速记·巧练(苏教版2019必修第二册)(已下线)12.3 复数的几何意义-【帮课堂】(苏教版2019必修第二册)(已下线)第七章 复数(提升卷)--重难点突破及混淆易错规避(人教A版2019必修第二册)上海市嘉定区、长宁、金山区2019-2020学年高三上学期期末数学试题2020届上海市长宁嘉定金山高三一模数学试题2020届上海市嘉定区高三一模数学试题沪教版(2020) 必修第二册 领航者 第9章 复数 9.2复数的几何意义 第1课时 复平面与复数的坐标、向量表示及复数加法的平行四边形法则沪教版(2020) 必修第二册 同步跟踪练习 第9章 复数 单元测试卷河北省石家庄市藁城新冀明中学2020-2021学年高一下学期(5月)第二次月考数学试题沪教版(2020) 必修第二册 领航者 一课一练 第9章 9.2 第1课时 复平面与复数的坐标、向量表示及复数加法的平行四边形法则(已下线)12.3-4 复数的几何意义、三角表示-2021-2022学年高一数学10分钟课前预习练(苏教版2019必修第二册)沪教版(2020) 必修第二册 单元训练 第9章 单元测试(B卷)沪教版(2020) 必修第二册 同步跟踪练习 第9章 测试卷重庆市缙云教育联盟2023-2024学年高一下学期3月月度质量检测数学试题
名校
解题方法
6 . 复数是由意大利米兰学者卡当在十六世纪首次引入,经过达朗贝尔、棣莫弗、欧拉、高斯等人的工作,此概念逐渐为数学家所接受.形如
的数称为复数,其中
称为实部,
称为虚部,i称为虚数单位,
.当
时,
为实数;当
且时,
为纯虚数.其中
,叫做复数
的模.设
,
,
,
,
,
,
如图,点
,复数
可用点
表示,这个建立了直角坐标系来表示复数的平面叫做复平面,
轴叫做实轴,
轴叫做虚轴.显然,实轴上的点都表示实数;除了原点外,虚轴上的点都表示纯虚数.按照这种表示方法,每一个复数,有复平面内唯一的一个点和它对应,反过来,复平面内的每一个点,有唯一的一个复数和它对应.一般地,任何一个复数
都可以表示成
的形式,即
,其中
为复数
的模,
叫做复数
的辐角,我们规定
范围内的辐角
的值为辐角的主值,记作
.
叫做复数
的三角形式.
,
,求
、
的三角形式;
(2)设复数
,
,其中
,求
;
(3)在
中,已知
、
、
为三个内角
的对应边.借助平面直角坐标系及阅读材料中所给复数相关内容,证明:
①
;
②
,
,
.
注意:使用复数以外的方法证明不给分.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3d2c0c72c17b74f9a5a175ec2b9d77e7.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0a6936d370d6a238a608ca56f87198de.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2c94bb12cee76221e13f9ef955b0aab1.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c789a7cd7ac2b8b96dc879c6c8161ee4.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/143b917df0520097be222accbddf9394.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e81e59019989b7dc2fb59b037ef6e010.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b03b011f69dfc5262a3d82f64676739b.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e81e59019989b7dc2fb59b037ef6e010.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/22fe68ea0bf368925909606949da47f3.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e81e59019989b7dc2fb59b037ef6e010.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5a0bf9b2a7378e73e9fd06c693bfda07.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/368f9e12546277731776041c73dbe58c.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0a6936d370d6a238a608ca56f87198de.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2c94bb12cee76221e13f9ef955b0aab1.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/071a7e733d466949ac935b4b8ee8d183.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/80e80e5baee553150c67a91f1017a7be.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/eb6958203312cbda12fd2683a819dd9d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e472aea001d179c284e3687a9aacf384.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4b40b6895776e0807c2baecbc8f33a8c.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e472aea001d179c284e3687a9aacf384.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/81dea63b8ce3e51adf66cf7b9982a248.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d053b14c8588eee2acbbe44fc37a6886.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4b40b6895776e0807c2baecbc8f33a8c.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ec45476379fb51aa1ef0a93f849f48be.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1e283f3168c0b5e8f68dda92c43651e6.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/11bc05f41215f9894e11d1df0465751a.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e81e59019989b7dc2fb59b037ef6e010.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c24095e409b025db711f14be783a406c.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e81e59019989b7dc2fb59b037ef6e010.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8eec3e684af41f9ed4db5b931b9ccfb6.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c24095e409b025db711f14be783a406c.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8f56cfb41ee7cb758fee138ab09e0d13.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ec45476379fb51aa1ef0a93f849f48be.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4b40b6895776e0807c2baecbc8f33a8c.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3665b2dac544bfb2a0c175f95a480e1e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/91b3a15906b84b98a3ac563e7e2ec9c3.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6fe615164ed2995bdeea0f5b0ba94231.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ec04f844e8fd9d9b1ef835e23eaa54e2.png)
(2)设复数
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/fd87d6e1987cf95d102de1045d3722a7.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/398d8980d3ec9fbf536a1efa6312a19a.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e0492634f27279b6470798af0185be67.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9c723970ac738976e0130e1438b67058.png)
(3)在
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/15c0dbe3c080c4c4636c64803e5c1f76.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0a6936d370d6a238a608ca56f87198de.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2c94bb12cee76221e13f9ef955b0aab1.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/071a7e733d466949ac935b4b8ee8d183.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/24e0c10fb103930eabd5fa18e8f9bb06.png)
①
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1501d4035822b34fcc2378f1e316f159.png)
②
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e63471f592531e46277365ed319e2acc.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b923694c299d953e02cb79dfcef9f56a.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a5ce2f54d69a5987c1de19da53342811.png)
注意:使用复数以外的方法证明不给分.
您最近一年使用:0次
2024-03-12更新
|
583次组卷
|
4卷引用:第七章:复数(新题型)-同步精品课堂(人教A版2019必修第二册)
(已下线)第七章:复数(新题型)-同步精品课堂(人教A版2019必修第二册)黑龙江省哈尔滨师范大学附属中学2023-2024学年高一下学期开学考试数学试卷重庆市缙云教育联盟2023-2024学年高一下学期3月月度质量检测数学试题(已下线)模块五 专题六 全真拔高模拟2
7 . 对于非空集合
,定义其在某一运算(统称乘法)“×”下的代数结构称为“群”
,简记为
.而判断
是否为一个群,需验证以下三点:
1.(封闭性)对于规定的“×”运算,对任意
,都须满足
;
2.(结合律)对于规定的“×”运算,对任意
,都须满足
;
3.(恒等元)存在
,使得对任意
,
;
4.(逆的存在性)对任意
,都存在
,使得
.
记群
所含的元素个数为
,则群
也称作“
阶群”.若群
的“×”运算满足交换律,即对任意
,
,我们称
为一个阿贝尔群(或交换群).
(1)证明:所有实数在普通加法运算下构成群
;
(2)记
为所有模长为1的复数构成的集合,请找出一个合适的“×”运算使得
在该运算下构成一个群
,并说明理由;
(3)所有阶数小于等于四的群
是否都是阿贝尔群?请说明理由.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/38bdb8d4c486c37ac64517ed8d60888b.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a1240721fdf6d8e1ed9c1158ae723637.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7c88c74a9c2efc329f92aa4203f0f780.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7c88c74a9c2efc329f92aa4203f0f780.png)
1.(封闭性)对于规定的“×”运算,对任意
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0c9cbad1e8b405feac6e8fe403f024b8.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/830afd1befcf1a92874b5e0bc214578d.png)
2.(结合律)对于规定的“×”运算,对任意
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0d4ef2c168b3dba086f2485c3c9cc7c5.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/44d2d88c195317bf5827a1304068f26a.png)
3.(恒等元)存在
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/25e6faeeed98a19d7012c921ca71a046.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/572f76c63e3a74a90a1e6ca5ae401cf8.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/96e4c22a6a498e197149ce29d9e98fce.png)
4.(逆的存在性)对任意
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/572f76c63e3a74a90a1e6ca5ae401cf8.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f5463eaf01a62bc6a772301d9e2ad19c.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1487cef1d4227621d9311541dec87156.png)
记群
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7c88c74a9c2efc329f92aa4203f0f780.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f9f2eb65f2fe6546a5e318343d25fe66.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7c88c74a9c2efc329f92aa4203f0f780.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f9f2eb65f2fe6546a5e318343d25fe66.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7c88c74a9c2efc329f92aa4203f0f780.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9d22c891ccf3768b616c5ddaad575aca.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8c679fe86736064c65a292db59cb739c.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7c88c74a9c2efc329f92aa4203f0f780.png)
(1)证明:所有实数在普通加法运算下构成群
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/74ba4c1c25324a8875b09d0c855a82ad.png)
(2)记
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d719c0c82f38a886db86c71bbaf8db32.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d719c0c82f38a886db86c71bbaf8db32.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/de4ef5fbf807246591e03d07ba4e3a4e.png)
(3)所有阶数小于等于四的群
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7c88c74a9c2efc329f92aa4203f0f780.png)
您最近一年使用:0次
2024高三·全国·专题练习
8 . 设
,
,
,
,
为
个复数.
(1)如果
,求证:
;
(2)若
,则有什么样的结果?
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/af68f652b4c13657ffddf3c9e7eb262b.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/fa224ed9be8766a4d0b5138bd57de0f0.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0a67a742d2a43e907fb1c3a1bdf1d6a9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/daa5e9bd516f6282483b92cfe6074623.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d9dc4e868a310c371ff88075d8a966a6.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b6a24198bd04c29321ae5dc5a28fe421.png)
(1)如果
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/beddfb49ec7b9eb6e3b83808b99ffbe6.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8bc11168f926631ddf7fe19b6cda4896.png)
(2)若
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7cc73be63fd24c42c75475bed0451a33.png)
您最近一年使用:0次
9 . 数学中的数,除了实数、复数之外,还有四元数.四元数在计算机图形学中有广泛应用,主要用于描述空间中的旋转.集合
中的元素
称为四元数,其中i,j,k都是虚数单位,d称为
的实部,
称为
的虚部.两个四元数之间的加法定义为![](https://staticzujuan.xkw.com/quesimg/Upload/formula/de91529a789f974a5d24401d6055271d.png)
.
两个四元数的乘法定义为:
,四元数的乘法具有结合律,且乘法对加法有分配律.对于四元数
,若存在四元数
使得
,称
是
的逆,记为
.实部为0的四元数称为纯四元数,把纯四元数的全体记为W.
(1)设
,四元数
.记
表示
的共轭四元数.
(i)计算
;
(ii)若
,求
;
(iii)若
,证明:
;
(2)在空间直角坐标系中,把空间向量
与纯四元数
看作同一个数学对象.设
.
(i)证明:
;
(ii)若
是平面X内的两个不共线向量,证明:
是X的一个法向量.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/503f295e33e64c58837fbffe80d50ee6.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/60ad8ba38041748888882382aafc53e5.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e170f206fdbbd834aad7580c727e2cc6.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6a143dc52a9036a83bdf6d30b56d8269.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e170f206fdbbd834aad7580c727e2cc6.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/de91529a789f974a5d24401d6055271d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e515963c8bd254633208aff7645abec9.png)
两个四元数的乘法定义为:
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/40e08a9f609ec5961b2d60416b816c10.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e170f206fdbbd834aad7580c727e2cc6.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5b5858ee1ce52b251816757257a11c29.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4f1d8088a83d194f555095e667019f04.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5b5858ee1ce52b251816757257a11c29.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e170f206fdbbd834aad7580c727e2cc6.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/026e0d7943bcddc8c8ba91757b4186d5.png)
(1)设
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d3b40d453ac56a449af2c33e31ff49c5.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/60ad8ba38041748888882382aafc53e5.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f2e4915f7ea4c5adb116410a2aa0c3bd.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e170f206fdbbd834aad7580c727e2cc6.png)
(i)计算
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a62de470f4c58383a0c963372924b618.png)
(ii)若
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ea4313f830e9be762a14205f2c2141d7.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f158b589206bf9741a1802a4d2a8fb8b.png)
(iii)若
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c3704cc0a9865a91a680228e2f0aa6ca.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4e80518e5dbd2ce5243e9f043021f33d.png)
(2)在空间直角坐标系中,把空间向量
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/280473bf8b2088551dd608fb60ff4866.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3002354512a65ed4963ee04ef1801d8b.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/fee4659ae1953845093516fef650d281.png)
(i)证明:
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c1a6a1f8cd81e048b47ae4ca5a88f727.png)
(ii)若
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c4e288596fa3811dd2c17bded60e82e7.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9f435efcc7869eec21bdba1ed81dc3f5.png)
您最近一年使用:0次
10 . 通过平面直角坐标系,我们可以用有序实数对表示向量.类似的,我们可以把有序复数对
看作一个向量,记
,则称
为复向量.类比平面向量的相关运算法则,对于
,
,
、
、
、
、
,我们有如下运算法则:
①
; ②
;
③
; ④
.
(1)设
,
,求
和
.
(2)由平面向量的数量积满足的运算律,我们类比得到复向量的相关结论:
①![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1cb72256695bffefefffc1572fc08f45.png)
②
③
.
试判断这三个结论是否正确,并对正确的结论予以证明.
(3)若
,集合
,
.对于任意的
,求出满足条件
的
,并将此时的
记为
,证明对任意的
,不等式
恒成立.
根据对上述问题的解答过程,试写出一个一般性的命题(不需要证明).
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/bbee4027127a0bce1cdc3fc50d28c5dc.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/bd1d1ef701f3618fa1884a3791d366aa.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7a2f4b1178f68bd147d1a2a6acd04435.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/bd1d1ef701f3618fa1884a3791d366aa.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/aaa0a749b475d60688fac80c38156eea.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/af68f652b4c13657ffddf3c9e7eb262b.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/fa224ed9be8766a4d0b5138bd57de0f0.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0a67a742d2a43e907fb1c3a1bdf1d6a9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3b29a77cfdb8d2a0b684389921e1496c.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d7def0e6fc765f99565eaa1d498e291c.png)
①
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/aaaebd6ed5e92ec8986cbe043ab574ef.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c5b8ba665154ad6f7ccb8ca422837e7c.png)
③
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/bcd637dcc0c2703912c91ad32bbd7dc0.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/fcc6b415aea966f160e3f3085cef1f6e.png)
(1)设
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/fad6cc9ce836150c84f3c7b354e15057.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/58b017a79eadd64416f98c7acb0f5bdd.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d9cd8bbf47b69bbd7a6263b041290d11.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c39d1d88189726ae99c309644fca3494.png)
(2)由平面向量的数量积满足的运算律,我们类比得到复向量的相关结论:
①
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1cb72256695bffefefffc1572fc08f45.png)
②
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4b55031cf0985ff92dd0c16f1ad4d01b.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8d7e78abccf1d9228fdf68e7ecf58465.png)
试判断这三个结论是否正确,并对正确的结论予以证明.
(3)若
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7f943fd00c91acee53d2e9f4b31a5437.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/cb8ed19b9d61c48d77a9fc37335f47f1.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0d1e98efe26c2c1442f6a73f09ec8d01.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/40891013fa6a2a7ccee812efe7643e04.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a5b6dbee41d492940e58103a9aaa2e7d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c94075193c11fe43f2396cff5a485054.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c94075193c11fe43f2396cff5a485054.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1b452962126ea36badc6354f5e2b1d80.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0d1e98efe26c2c1442f6a73f09ec8d01.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b8ab37842e5e918ff46a4089e234d04b.png)
根据对上述问题的解答过程,试写出一个一般性的命题(不需要证明).
您最近一年使用:0次
2023-07-06更新
|
557次组卷
|
7卷引用:专题7.4 复数运算的综合应用大题专项训练-举一反三系列-
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