# [Special Relativity] - Finding the Proper Length at Rest in Frame S'

Homework Statement:
A rod of proper length ##l_0## is at rest in a frame ##S'##. It lies in the ##(x', y')## plane and makes an angle of ##l=\sqrt{2}c## with the ##x'## axis. If ##S'## moves with constant velocity ##v## parallel to the ##x## axis of another frame ##S##:

a. What must be the value of ##v## if, as measured in ##S##, the rod is at 45 degrees to the ##x## axis?

b. What is the length of the rod as measured in ##S## under these conditions?
Relevant Equations:
$$\gamma = \frac{1}{\sqrt{1- \frac{v^2}{c^2}}}$$
Currently, the only part of the textbook question that is completely throwing me off is "an angle of ##l=\sqrt{2}c##". If I am not mistaken, how am I suppose to interpret that as an angle and calculate for the answers of (a) and (b) accordingly?

As for my attempted solution process of this question, please refer to the below link (problem 3).

https://www.atmosp.physics.utoronto.ca/people/strong/phy140/tut12ans_01.pdf

The above link takes the solution approach I would normally take with this problem if they provided an actual angle (e.g. ##\theta = 45## degrees). Ironically enough (despite minor differences in the question), the answers in the above PDF happens to be the exact same as my above question (according to the textbook).

That said, the only aspect of this problem confusing me is - as mentioned - the part of the text that includes ##l=\sqrt{2}c## as the angle. How should I interpret this and include it as part of my calculation?

Thank you for your help!

PeroK