是否存在三个垂直向量
空间是否存在三个整数坐标相互垂直的向量。且9个坐标整数最多只有一个是0. 为了找三个整数坐标向量垂直。貌似要是有4个数为0，结果肯定一大堆。全部数非0，还不知道有没有解其实我还可以将这个问题化成只有三个角度函数的三角函数表达式，只是要满足所有坐标是整数会更难解 还有三个向量不一样长的情况，只相互垂直，这样有没有整数解，还是不许结果0太多了 随便用几个有理的平面旋转矩阵乘乘就出来了
{{52, 39, 0}, {36, -48, 25}, {15, -20, -60}} 终极问题就是a^2+b^2+c^2=x^2+y^2+z^2=1,ax+by+cz=0.求有理数解。不要有0的解 简化一次：a,b,z是非零有理数，(1-a^2-b^2)^(1/2)与(a^2+b^2-z^2)^(1/2)也是非零有理数。求a,b,z 简化一次：a,b,z是非零有理数，(1-a^2-b^2)^(1/2)与(a^2+b^2-z^2)^(1/2)也是非零有理数。求a,b,z 有一个零的是两个旋转矩阵乘积。三个矩阵乘积，除非恰好碰上和为零
Do]^-1 #&,Sin,0},{-Sin,Cos,0},{0,0,1}},{{Cos,0,Sin},{0,1,0},{-Sin,0,Cos}},{{1,0,0},{0,Cos,Sin},{0,-Sin,Cos}}}]]/.(#->ArcCos[(2m n)/(m^2+n^2)/.{m->RandomInteger[{1,10^2}],n->RandomInteger[{1,10^2}]}]&/@{a,b,c})]],{10}]
{{32787752160,30474825004,32401397397},{-41451697565,35534391360,8524435680},{-16134505920,-29363144853,43944119596}}
{{9217256412,13402361120,1677227355},{-13492884955,9228906960,404379612},{-615167280,-1611888237,16260921760}}
{{47309248800,52799257147,24666522804},{-58269456000,43365326004,18933645253},{-932416205,-31081358880,68318691840}}
{{165311971200,198053717220,59998455335},{-203719298020,169247528079,2619011772},{-36380450265,-47782179772,257966012304}}
{{93234615,22245452,3785536560},{-3700600748,-797399265,95828580},{797706720,-3701774460,2106377}}
{{202552875,18682983000,367810300},{-17412536636,322200648,-6777161685},{-6781792248,-269256061,17411632920}}
{{1235865111,1287492148,479005380},{-1267723548,820755936,1064744785},{529111700,-1040753775,1432242000}}
{{409332240,96215971,742100472},{-249533195,815005872,31970604},{-705481260,-232446600,419271125}}
{{144606132,219011724,613661125},{-647341500,119805125,109785060},{-74128901,-618981132,238378500}}
{{540603960,859075139,195106548},{-790207355,371685048,552948336},{389421360,-438371340,851182045}} 你这找的数字好大啊，我找到一组没有相同数，也没有相反数（15,120,140）（-168，-49,60）（76，-132,105）
看来想知道限定范围空间中整数格点里取8个点构成立方体可以构成多少个立方体，这个问题难于上青天
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