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I'm working on a couple of problems regarding simultaneity.

For one I'm given events X_a = (ct,x,y,z) = (2,1,-3,2) and X_b = (6,3,-1,5)

I want to find out if there's a frame of reference where the events are simultaneous.

I find the change vector X_b - X_a = (Δct,Δx,Δy,Δz) = (4,2,2,3)

I quickly find out that it's impossible to have the events simultaneous with a pure boost in the x, y and z direction. So I used the Lorentz-transformation matrix for a boost in any direction I found on Wikipedia.

I get the following equation γ(4-2β_x - 2β_y - 3β_z) = 0 or 4c^2 = 2v_x + 2v_y + 3v_z

Here we obviously have 0 < β_i < 1 and |v_i| < c

So I can't exactly make one v_i a pivot and the rest free variables. Any ideas?