IBM Ponder This #080 · December 2004

Puzzle for December 2004

This month's puzzle is proposed by Michael Brand.

Consider N vectors, V_1 ... V_N, (not all zero) in an N-dimensional space. Project these vectors orthogonally 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.

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.

Solution

Best opened after a real attempt

To be added.