解题方法
1 . 已知向量
,且
与
的夹角为
.
(1)求证:(
;
(2)若
与
的夹角为
,求
的值.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ad59ee7969f2a082ed53bdf0aaa748ed.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9413091ff21dc5919dbf66497c552946.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/36fe4e713c108e118522a99ecd683924.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/286d0d390a6b3783f425eac558b118dc.png)
(1)求证:(
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/67d9527e5ff65a51c20eccba34e6a40d.png)
(2)若
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4f7a1df960feef63dec4790d63f52279.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d1b8a88a16125366536cb4ad658e0cf1.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ac1a63ab608517bb10aa036783dfb51f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/df64046e91b047037f19e4032e3b6de3.png)
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名校
2 . 如图,在
中,
.
为等边三角形.
(2)试问当
为何值时,
取得最小值?并求出最小值.
(3)求
的取值范围.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/15c0dbe3c080c4c4636c64803e5c1f76.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/98f6cab08aec875683b343184a701c7f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/15c0dbe3c080c4c4636c64803e5c1f76.png)
(2)试问当
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/df64046e91b047037f19e4032e3b6de3.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/68a116fa5c4c6d67aa3454ef87826718.png)
(3)求
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/50eae187c6532ec0abf6fd40b0cc2da6.png)
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3卷引用:山东省聊城第一中学等部分学校2023-2024学年高一下学期5月质量监测联合调考数学试题
解题方法
3 . 如图,
分别是等腰梯形
的边
上的动点,
,其中
为定值,
,设
,其中
.
,求出
的表达式;
(2)证明:
的余弦值与
的取值无关;
(3)求
的取值范围.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f6bce3d91ca23b86d8c6625f2632e437.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/411b38a18046fea8e9fab1f9f9b80a5f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4ced9844fe2e052c70486af0740afa63.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/596d41b6556f383445536d1c534ac182.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a1f07948e9258b482a2164ac871f90f9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7a18fec3e4a4fbc7b3e0e037ce650023.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4d91513d2e546a5a0b5fd42379db8df2.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/94e72cf7374a65ced433b6fa113ef57d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/15e2be3e9225c71609248299caa49432.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/192c2f1059f6e05d44df048f5fdca04b.png)
(2)证明:
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4ff7023ec0f513c7d0ef86859a5ede54.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/294f5ba74cdf695fc9a8a8e52f421328.png)
(3)求
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5db625151987f893816de66b15d9e699.png)
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解题方法
4 . 已知双曲线
:
(
,
)的右焦点为
,一条渐近线的倾斜角为60°,且
上的点到
的距离的最小值为1.
(1)求
的方程;
(2)设点
,
,动直线
:
与
的右支相交于不同两点
,
,且
,过点
作
,
为垂足,证明:动点
在定圆上,并求该圆的方程.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c5db41a1f31d6baee7c69990811edb9f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/19f3fa0b40fb0d9b8c62e37316ab3b04.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/94440d3e4c073f94f2b266ff99d50e74.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/67ca5fd57c2c2fcc3c7a574fdd1467d9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a0ed1ec316bc54c37c4286c208f55667.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c5db41a1f31d6baee7c69990811edb9f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a0ed1ec316bc54c37c4286c208f55667.png)
(1)求
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c5db41a1f31d6baee7c69990811edb9f.png)
(2)设点
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/19b62194097ac66a5093c57fca2f5b4f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/15ed90ebf0061c8a79beed307fc1719a.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0f85fca60a11e1af2bf50138d0e3fe62.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/15b256345d7109e081b7c895591e995d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c5db41a1f31d6baee7c69990811edb9f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5963abe8f421bd99a2aaa94831a951e9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7f9e8449aad35c5d840a3395ea86df6d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b3900a0379f92c09ee1d9a5baa5dc362.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1dde8112e8eb968fd042418dd632759e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/abf59107b1cb4f2c9dc9f27525311e22.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/73465a1f9aa03481295bf6bd3c6903ac.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/73465a1f9aa03481295bf6bd3c6903ac.png)
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2023-03-24更新
|
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2卷引用:山东省聊城市2023届高三下学期第一次模拟数学试题
名校
解题方法
5 . 已知梯形
中,
,
,
,E为
的中点,连接AE.
(1)若
,求证:B,F,D三点共线;
(2)求
与
所成角的余弦值;
(3)若P为以B为圆心、BA为半径的圆弧
(包含A,C)上的任意一点,当点
在圆弧
(包含A,C)上运动时,求
的最小值.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/411b38a18046fea8e9fab1f9f9b80a5f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f864244952b60f3648f08a19268efae9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f121eabff3c62c1a196d9ca5f6f83f0b.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e075468e7fb0bf30229aec01a7205977.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0dc5c9827dfd0be5a9c85962d6ccbfb1.png)
(1)若
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9575824984c3e936744641879dc3edd4.png)
(2)求
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6d021a5c98388463d577675e58068aa7.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4781e3daa2c4e018ca0ae09bb56abc0f.png)
(3)若P为以B为圆心、BA为半径的圆弧
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/667349d99185bb045030b733352ff7fd.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/dad2a36927223bd70f426ba06aea4b45.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/667349d99185bb045030b733352ff7fd.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ab7aaa871ceb78e5b80b531a7cf4f1c9.png)
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2023-03-26更新
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4卷引用:江苏省常州市联盟学校2022-2023学年高一下学期3月学情调研数学试题
6 . 如图,正方体
的棱长为a.
和
的夹角;
(2)求证:
.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b3da8c338342e38c9aa3f274c053fd5b.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e663220a66eff19da6a71e46b397db2e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/020ebe1219437129358b986eb9e70bbf.png)
(2)求证:
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8c702d582ef87c9423724dcb579b6e2d.png)
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4卷引用:河南省周口市太康县第二高级中学2022-2023学年高二上学期12月月考文数试题
河南省周口市太康县第二高级中学2022-2023学年高二上学期12月月考文数试题河南省周口市太康县第二高级中学2022-2023学年高二上学期12月月考理数试题(已下线)第02讲 1.1.2空间向量的数量积运算(7类热点题型讲练)-【帮课堂】2023-2024学年高二数学同步学与练(人教A版2019选择性必修第一册)人教A版(2019)选择性必修第一册课本习题 习题1.1
7 . 已知
是两个单位向量,
,且
.
(1)求
的夹角;
(2)若D为线段BC上一点,DC =2BD,求证:AD⊥AB.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1238a7a8ce9db1a47ae0ac80d5ed70cf.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/611ab4e3ede21cd03d0cf99675d9ed22.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ce69b7bf1677649ee5121f8a884f6f44.png)
(1)求
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1238a7a8ce9db1a47ae0ac80d5ed70cf.png)
(2)若D为线段BC上一点,DC =2BD,求证:AD⊥AB.
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8 . (1)已知
,
是两个不平行的向量,向量
,
,
,求证:A,C,D三点共线;
(2)已知
,
满足
,
,
,求
.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7a2f4b1178f68bd147d1a2a6acd04435.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c94075193c11fe43f2396cff5a485054.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4630c9846ba7d1dd7946b0dcf3a20a09.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/701419d3bc35616c246c2539211cfee0.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a4487b5e0b22ec295eb92ab8a1b39a8a.png)
(2)已知
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7a2f4b1178f68bd147d1a2a6acd04435.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c94075193c11fe43f2396cff5a485054.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/70e310702748855c3edccd151aae0143.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b113dc271cf51b3018bd1de14edf73ad.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6c97632424d6ca33502e84a20bc74890.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d8403f831ff29aff5759ebdccc671377.png)
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解题方法
9 . 在
中,
,对任意
,有
.
(1)求角
;
(2)若
,
,且
、
相交于点
.求证:
.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/15c0dbe3c080c4c4636c64803e5c1f76.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/883fc5e3faf39829d60804b59deb1730.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c4166972dec0aa3e8694a44eeb941a08.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2b72a248f96602a5b6de57929686c56b.png)
(1)求角
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5963abe8f421bd99a2aaa94831a951e9.png)
(2)若
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3d2997d56e933559b51ff87ac8d07e60.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5dcf2f445f03af259458d5f4478f7ffa.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/85c4bdfb0db1e31e8459df1d15f9ab55.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9d78abbad68bbbf12af10cd40ef4c353.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/dad2a36927223bd70f426ba06aea4b45.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/12319a1cdcc58d25c30d2b3ab5848237.png)
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10 . 设
是两个夹角为
的单位向量,
.
(1)证明:A,B,D三点共线
(2)若
,且
与
所成角为
,求实数k的值.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/111939547e581e8bf029a241b9e9cb05.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/be6a6301878fed2a01413020b27310a5.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/dba1d184ce4ed77c3bb1d3375508cec6.png)
(1)证明:A,B,D三点共线
(2)若
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/63eb66cbc4416a489145eb2456f9c6d7.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f75fc81977aee721525b4c5625f5a097.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c3ed9c2e9fabbfc63733bae8fa079d00.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f6b86c22b670a8e9f3896f9e8883fbbb.png)
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