Re: [Algorithms] decompose onto non-orthogonal vectors

[<ro...@do...>]

Peter Dimov  wrote:

>> >> how to you get scalars (u,v) such that u*a + v*b = p?
>> >> Ie. decompose p onto a and b.
>> >
>> >Easy. dot the equation with a and b, correspondingly:
>> >
>> >u * (a dot a) + v * (b dot a) = p dot a;
>> >v * (a dot b) + v * (b dot b) = p dot b.
>> >
>> >Solve the system.
>>
>> In the 2D case this is more complicated than it need be, but will give
>> the correct solution (provided, of course that one knows dot product
>> and how to solve a 2x2 system).
>>
>> In the 3D case, this will give the correct solution only if p is
>> indeed in the plane of a and b.  It will not detect that there is no
>> solution if p is not coplanar with a and b.
>
>This works for any number of dimensions. Whenever a solution exists, the (u,
>v) pair determined by this approach will have the property u * a + v * b =
>p. (Easy to prove.)

Agree that it works for any number of dimensions provided that p does
indeed lie in the plane of a and b.  My point was that it will not
detect the case that p does not lie in that plane, that is, will not
detect the case that no solution exisits.