2018高三上·全国·专题练习
1 . 下列说法中:
①两个有共同起点且相等的向量,其终点一定相同;
②若
,则
;
③若非零向量
,
共线,则
;
④若向量
,则向量
,
共线;
⑤由于零向量的方向不确定,故其不能与任何向量平行;
其中正确的序号为_______________________ .
①两个有共同起点且相等的向量,其终点一定相同;
②若
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ab14f1db85f43583c37c416c1bcacd67.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e4b7ca4667865ebe617fa6539e8d11fd.png)
③若非零向量
![](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/e4b7ca4667865ebe617fa6539e8d11fd.png)
④若向量
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e4b7ca4667865ebe617fa6539e8d11fd.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/64c5562bd4d1b54424330cb6329cd79d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b45ba716f03748c19b7ce2f99af536ab.png)
⑤由于零向量的方向不确定,故其不能与任何向量平行;
其中正确的序号为
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2019高三·全国·专题练习
2 . 若非零向量
与
互为相反向量,给出下列结论:①
∥
;②
≠b;③|
|≠|
|;④
=-
.其中所有正确命题的序号为____ .
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f45ea1d522a3ebbf5f5d8d330fdbd1e6.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/87b549e16842d85432dfb178dc0de58e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f45ea1d522a3ebbf5f5d8d330fdbd1e6.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/87b549e16842d85432dfb178dc0de58e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f45ea1d522a3ebbf5f5d8d330fdbd1e6.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f45ea1d522a3ebbf5f5d8d330fdbd1e6.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/87b549e16842d85432dfb178dc0de58e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/87b549e16842d85432dfb178dc0de58e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f45ea1d522a3ebbf5f5d8d330fdbd1e6.png)
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3 . 在推导很多三角恒等变换公式时,我们可以利用平面向量的有关知识来研究,在一定程度上可以简化推理过程.如我们就可以利用平面向量来推导两角差的余弦公式:
.具体过程如下:
如图,在平面直角坐标系
内作单位圆
,以
为始边作角
,
.它们的终边与单位圆
的交点分别为A,B.
![](https://img.xkw.com/dksih/QBM/editorImg/2023/1/14/ff7914f9-3736-4a49-b171-4c14d5491d7e.png?resizew=342)
则
,
,由向量数量积的坐标表示,有
.
设
,
的夹角为
,则
,
另一方面,由图(1)可知,
;
由图(2)可知
,于是
,
.
所以
,也有
;
所以,对于任意角
,
有:
.
此公式给出了任意角
,
的正弦、余弦值与其差角
的余弦值之间的关系,称为差角的余弦公式,简记作
.有了公式
以后,我们只要知道
,
,
,
的值,就可以求得
的值了.
阅读以上材料,利用图(3)单位圆及相关数据(图中M是AB的中点),采取类似方法(用其他方法解答正确同等给分)
解决下列问题:
![](https://img.xkw.com/dksih/QBM/editorImg/2023/1/14/2652d775-a20f-41fd-944a-9d388f0b4a1d.png?resizew=274)
(1)判断
是否正确?(回答“正确”,“不正确”,不需要证明)
(2)证明:
.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5ff8eb79da2ae1202feebf45ba5e795c.png)
如图,在平面直角坐标系
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7ee31829d0d4d5f779a957d7df8058ab.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1dde8112e8eb968fd042418dd632759e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c3e5af20b2f8c1fba4470f9650989e51.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/1dde8112e8eb968fd042418dd632759e.png)
![](https://img.xkw.com/dksih/QBM/editorImg/2023/1/14/ff7914f9-3736-4a49-b171-4c14d5491d7e.png?resizew=342)
则
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/92ab3f6bd216fc240a107a8dd7e1acdd.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/af35399a864361859b2fc9abe4471a85.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/437ebce60a1d755209353f0d94462154.png)
设
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/588414d07bcedbf1e7d46d0d028e269d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5773af927ab0caa208eef1adf9e87aee.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c24095e409b025db711f14be783a406c.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c5351800b6c0891ab2946d1ccd2f6c2d.png)
另一方面,由图(1)可知,
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/655ee7e11f540619722504916419e009.png)
由图(2)可知
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/18eedcc65589e7529da85a578bd0ecb2.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c24483522263bb3d2c4275c993ef542e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/45a173784888adf2946382fa093ba53a.png)
所以
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2084f038effd4b810eb59e6a9942684d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5ff8eb79da2ae1202feebf45ba5e795c.png)
所以,对于任意角
![](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/3cd78eb0780bb4395457cc463763991d.png)
此公式给出了任意角
![](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/fd927b4b5a7875528c1b54aa4bb8b2dd.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7c5bcf44b6a1dd4daf8eca077ff72d4c.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7c5bcf44b6a1dd4daf8eca077ff72d4c.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4d66c03d4ca06819a6ce7fc8ea6de0f0.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3eacde1c42151734fdc60f3001b590de.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0b4179e1ab8705cf19ea7aaf48888843.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/63fe57d4fbae536de2e641d9d349fcf1.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/bbb006ea697b63a914eb487073f0abe1.png)
阅读以上材料,利用图(3)单位圆及相关数据(图中M是AB的中点),采取类似方法(用其他方法解答正确同等给分)
解决下列问题:
![](https://img.xkw.com/dksih/QBM/editorImg/2023/1/14/2652d775-a20f-41fd-944a-9d388f0b4a1d.png?resizew=274)
(1)判断
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/90f93aa4ff886e380c9b7c05dbafd08d.png)
(2)证明:
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1d3ee14a51561c0eae1c74153cc76866.png)
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