Special Relativity Particle Distance Question

[bon]

Homework Statement

In a given inertial frame S, two particles are shot out from a point in orthogonal directions with equal speeds v. At what rate does the distance between the particles increase in S?

Homework Equations

The Attempt at a Solution

Ok so i want to write the trajectories of the two particles in terms of 4 displacements. Then find the difference. Then find the rate of change

So X1 = (ct, x1, y1, z1)

X2 = (ct, x2, y2, z2)

The difference is (0,x1 - x2, y1 - y2, z1-z2)

The rate of chance is

(0, dx1/dt - dx2/dt, ...) But how do i simplify this using the fact they're orthogonal..?

Should i not solve in such a general way. better to pick two 4-displacement vectors obviously orthogonal. How would i do that?

I see that something like (ct, 0, 0, 0) is orthogonal to (0, vt, 0, 0) But these aren't the trajectories of two moving bodies in the inertial frame. Actually, the second one doesn't even have a time entry. Is this allowed?

 

[vela]

You're only dealing with one inertial reference frame, S, so relativity doesn't even enter into this problem. For simplicity, just assume one object moves in the y direction and the other one moves in the z direction. At time t, both objects will be a distance vt from the origin, one on the y-axis and one on the z-axis. What's the distance between them? Differentiate this expression with respect to time to find the rate of change of the distance.