Length of a Stick Moving at Speed v in an Angle

[Tony11235]

Prove that a stick of proper length L has a length L' in a frame in which it moves with speed v along a line that makes an angle theta with it's length is given by

$$L' = L \sqrt{\frac{1-v^2 / c^2}{1 - (v^2 / c^2)\sin^2{\theta}}}$$

My problem here is the picture I think. So the stick overall is moving with speed v, but the stick is not necessarily parallel to v, but at an angle created by the direction of v and the stick?

 

[quantumdude]

So the stick overall is moving with speed v, but the stick is not necessarily parallel to v, but at an angle created by the direction of v and the stick?

Yes.

You need to remember that your stick is only contracted in the direction of motion. So just for definiteness say that $\vec{v}=v\hat{i}$. Then the stick is contracted in the x-direction but not in the y-direction.

Once you have the two components in the frame in which the stick has velocity $\vec{v}$ you can find its total length.

 

[Tony11235]

Got it. I went that direction earlier, but for some reason it didn't look as if the expression I derived was equivalent. BUT after some algebra...