Consider N vectors, V_1 ... V_N, (not all zero) in an N-dimensional space. Project these vectors to a 2-dimensional plane, P. Let us denote the results for the i'th vector as (a_i,b_i) in Cartesian coordinates defined on the plane. We will be interested in considering these coordinates as a complex number: x_i = a_i + j*b_i.(adsbygoogle = window.adsbygoogle || []).push({});

Now, these N vectors hold the following property: regardless of the chosen P, the sum of all {x_i}^2 is always equal to 0.

The question: give an explicit description of the property that the N vectors most hold. Prove that this property is both sufficient and necessary.

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# Vectors property

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