高等数学-向量代数与空间解析几何习题

高等数学-向量代数与空间解析几何习题

向量概念的考察

向量

例1
image-20200905184450689
image-20200905184511797

或者根据交点到两边的对称点的两个向量是两个大小相等、方向相反的向量来做。

夹角余弦

例1
image-20200905174929818
image-20200905174845185
例2
image-20200906093626296
image-20200906093637707

向量的数量积

例1
image-20200905233247357
image-20200905233302354

向量的向量积

例1
image-20200906094519793
image-20200906094504383

向量的混合积

例1
image-20200906103157596
image-20200906103051483

两直线共面，混合积为0

空间平面

例1
image-20200905175405890
image-20200905175426305
image-20200905175444606
例2
image-20200905183510586
image-20200905183529769

此题也可用过直线的平面束的方法来做。

例3
image-20200905234542278
image-20200905234557874
例4
image-20200906080426002
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例5
image-20200906080806612
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例6
image-20200906081252618
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例7
image-20200906084306386
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例8
image-20200906094211792
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例9
image-20200906095723919
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例10
image-20200906100039278
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例11
image-20200906100646084
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例12
image-20200906100943223
image-20200906100953820

空间曲面

旋转曲面

旋转曲面的方程

例1
image-20200905175908197
image-20200905175854887
例2
image-20200905180815701
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例3
image-20200906110820088
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曲面某点处的法线

例1
image-20200906084953030
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例2
image-20200906090925529
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image-20200906090944633

曲面某点处的切平面

例1
image-20200906091426748
image-20200906091436624

空间直线

求空间直线

例1
image-20200905181526670
image-20200905181539751
例2
image-20200906101354857
image-20200906101406485

求直线在平面上的投影直线

一般用过直线的平面束的方法做

例1
image-20200905182432012
image-20200905182444098
例2
image-20200906101839101
image-20200906101851441

也可以先求出过直线L并垂直于平面\(\pi\)的平面\(\pi_2\)，则平面\(\pi\)与平面\(\pi_2\)的交线就是投影直线。

夹角

直线与平面的位置关系

例1
image-20200906092006813
image-20200906092017071
例2
image-20200906092930226
image-20200906092940261

两直线的夹角

例1
image-20200906092431414
image-20200906092413025
例2
image-20200906102311607
image-20200906102325137

两直线共面

参见向量的混合积。

两直线共面，混合积为0；两直线异面，混合积不为0.

距离

点到平面的距离

例1
image-20200906085321799
image-20200906085331720

平行平面之间的距离

例1
image-20200906085528863
image-20200906085536873

点到直线的距离

例1
image-20200905180407489
image-20200905180345790
例2
image-20200906090123300
image-20200906090134326

异面直线间的距离

例1
image-20200905185829684
image-20200905185845028
image-20200905185856732