1. From a boost in the $x$ direction, determine the form of a boost in a direction $\overrightarrow n=(\cos\theta, sin\theta, 0)$. Compare with the general form given in lecture.
2. Consider boosts $B_x(v)$ and $B_y(v)$ respectively in the $x$ and $y$ directions with the same value of the boost velocity $v$. Show that the product $B_x(v)B_y(v)$ is equivalent to a rotation of an angle $\phi$ around the $z$ axis, followed by a boost in a direction $\overrightarrow n=(\cos\theta, sin\theta, 0)$ of velocity $v_\theta$. Determine $\phi$, $\theta$ and $v_\theta$.
3. Show that the two boosts of question 2 do not commute. Is the commutator $[B_x(v),B_y(v)]=B_x(v)B_y(v)-B_y(v)B_x(v)$ a rotation?
4. What is the worldline $x(t)$ of a particle with a constant proper acceleration $\alpha$ in the $x$ direction, starting from rest at $t=0$?