### Related

##### orthogonal_complement

? orthogonal_complement;

2067;

Calculate

##### orthogonal_complement

? orthogonal_complement;

Calculate

##### orthogonal_complement

? orthogonal_complement;

2067;

Calculate

##### orthogonal_complement

? orthogonal_complement;

Calculate

### orthogonal_complement

Run Example
```(%i1)load  (eigen);
(%o1)           /usr/share/maxima/5.21.1/share/matrix/eigen.mac
(%i2) a  :  covect  ([1,  2,  3]);
[ 1 ]
[   ]
(%o2)                                [ 2 ]
[   ]
[ 3 ]
(%i3) b  :  covect  ([2,  1,  2]);
[ 2 ]
[   ]
(%o3)                                [ 1 ]
[   ]
[ 2 ]
(%i4) print  ("a  =  ",  a,  "  b  =  ",  b);
[ 1 ]          [ 2 ]
[   ]          [   ]
a  =   [ 2 ]   b  =   [ 1 ]
[   ]          [   ]
[ 3 ]          [ 2 ]
[ 2 ]
[   ]
(%o4)                                [ 1 ]
[   ]
[ 2 ]
(%i5) basis  :  orthogonal_complement  (a,  b);
[  1  ]
[     ]
(%o5)                            span([  4  ])
[     ]
[ - 3 ]
(%i6) first  (basis);
[  1  ]
[     ]
(%o6)                               [  4  ]
[     ]
[ - 3 ]
(%i7) ```
Run Example
```load  (eigen);
(%o1)           /usr/share/maxima/5.21.1/share/matrix/eigen.mac
(%i2) a  :  covect  ([1,  1,  0]);
[ 1 ]
[   ]
(%o2)                                [ 1 ]
[   ]
[ 0 ]
(%i3) b  :  covect  ([1,  0,  0]);
[ 1 ]
[   ]
(%o3)                                [ 0 ]
[   ]
[ 0 ]
(%i4) print  ("a  =  ",  a,  "  b  =  ",  b);
[ 1 ]          [ 1 ]
[   ]          [   ]
a  =   [ 1 ]   b  =   [ 0 ]
[   ]          [   ]
[ 0 ]          [ 0 ]
[ 1 ]
[   ]
(%o4)                                [ 0 ]
[   ]
[ 0 ]
(%i5) basis  :  orthogonal_complement  (a,  b);
[ 0 ]
[   ]
(%o5)                             span([ 0 ])
[   ]
[ 1 ]
(%i6) first  (basis);
[ 0 ]
[   ]
(%o6)                                [ 0 ]
[   ]
[ 1 ]
(%i7) ```
Run Example
```load (vector_rebuild);
(%o1)      /usr/share/maxima/5.21.1/share/vector/vector_rebuild.mac
(%i2) powerdisp : true;
(%o2)                                true
(%i3) v([args]) := covect(args);
(%o3)                      v([args]) := covect(args)
(%i4) a : v(1,2,3);
[ 1 ]
[   ]
(%o4)                                [ 2 ]
[   ]
[ 3 ]
(%i5) n : v(2, 25, 10);
[ 2  ]
[    ]
(%o5)                               [ 25 ]
[    ]
[ 10 ]
(%i6) oc : orthogonal_complement(n);
[ - 25 ]  [  0   ]
[      ]  [      ]
(%o6)                      span([  2   ], [  10  ])
[      ]  [      ]
[  0   ]  [ - 25 ]
(%i7) [b, c] : [first(oc), second(oc)];
[ - 25 ]  [  0   ]
[      ]  [      ]
(%o7)                        [[  2   ], [  10  ]]
[      ]  [      ]
[  0   ]  [ - 25 ]
(%i8) x : a + r * b + s * c;
[ 1 ]     [ - 25 ]     [  0   ]
[   ]     [      ]     [      ]
(%o8)                   [ 2 ] + r [  2   ] + s [  10  ]
[   ]     [      ]     [      ]
[ 3 ]     [  0   ]     [ - 25 ]
(%i9) ```

### Related Help

Help for Orthogonal_complement