Introduction:
In this scenario, we have two inertial frames of reference A and B, where B is moving along the x-axis with a velocity of 9×10^6 m/s with respect to frame A. We are given the coordinates of a rod in frame A, and we need to determine the coordinates of the same rod in frame B.

Understanding frames of reference:
Frames of reference are used to describe the position and motion of objects. An inertial frame of reference is a frame in which Newton's laws of motion hold true. In this case, frame A and frame B are both inertial frames.

Coordinates of the rod in frame A:
In frame A, the coordinates of the rod's two ends are given as (5, -5, 0). These coordinates represent the position of the rod in frame A's coordinate system.

Determining the coordinates of the rod in frame B:
To determine the coordinates of the rod in frame B, we need to account for the velocity of frame B with respect to frame A. Since frame B is moving along the x-axis with a velocity of 9×10^6 m/s, we can consider a Lorentz transformation to find the coordinates of the rod in frame B.

Lorentz transformation:
The Lorentz transformation equations relate the coordinates and time measured in one inertial frame to the coordinates and time measured in another inertial frame that is moving with a constant velocity relative to the first frame. In this case, we are interested in transforming the coordinates from frame A to frame B.

Coordinate transformation:
The Lorentz transformation equations for the x-coordinate can be written as:
x' = (x - vt) / sqrt(1 - (v/c)^2)

In this equation, x' represents the transformed x-coordinate in frame B, x represents the original x-coordinate in frame A, v represents the velocity of frame B with respect to frame A (9×10^6 m/s), t represents time, and c represents the speed of light (3×10^8 m/s).

Calculating the transformed x-coordinate:
Substituting the given values into the Lorentz transformation equation, we can calculate the transformed x-coordinate in frame B.

x' = (5 - (9×10^6)t) / sqrt(1 - ((9×10^6)/(3×10^8))^2)

Simplifying this equation will give us the x-coordinate of the rod in frame B as a function of time.

Conclusion:
By applying the Lorentz transformation equations, we can determine the coordinates of the rod in frame B. The transformed x-coordinate in frame B can be calculated using the given velocity and the Lorentz transformation equation for the x-coordinate. Remember to consider the time component as well, as the transformation equations involve both spatial and temporal coordinates.