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I have an equation system of the form Aix + Biy + Ciz = Di, where (x,y,z) is a unit vector, and (Ai, Bi, Ci, Di) are sets of measurements from a noisy system (with typically 3-5 independant readings).

My first intuition to solve this problem was to pose this as an overdetermined linear equation system AX = B where X = (x,y,z), and to solve for X. However, with that approach, I have no way to enforce that the solution for vector X is a unit vector.

Is there an elegant (or standard) solution to that problem, or should I simply dive into non-linear equation solving solutions?

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    Suggestion: If you have $n$ measurement sets, then you have $\binom{n}{3}=% \frac{n(n-1)(n-2)}{3!}$ ways of choosing $3$ of them and for each one solve the corresponding linear equation system $AX=B$. You could determine $% X^{\ast }=(x^{\ast },y^{\ast },z^{\ast })$ so that $\left\vert 1-\left( x^{\ast }\right) ^{2}-\left( y^{\ast }\right) ^{2}-\left( z^{\ast }\right) ^{2}\right\vert $ is minimum. Finally you would compute $X^{\ast \ast }=(x^{\ast \ast },y^{\ast \ast },z^{\ast \ast })=\frac{X^{\ast }}{\left\vert X^{\ast }\right\vert }$.2010-11-05

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