A rod of length 1m lies in the lab frame making angle θ=30 with the
x-axis. Find the length and the angle it makes with the x’-axis as seen by
an observer moving at u=0.8c along the y-direction.
The projection along the x axis, L*cos30, will not change due to relative motion, but the projection along the y axis will be shortened to
L*sin30*sqrt(1-(0.8)^2) = (L/2)*0.6 = 0.3 L
The new length is
L' = sqrt[(Lcos30)^2+(0.3L)^2]
= L*sqrt[.75 + .09] = 0.917 L
The new angle with the x axis is
arctan (0.3/0.866) = 19.1 degrees