Problem source

The position vectors, relative to an origin $O$,
at time $t$  of the particles $P$ and $Q$ are
$$\cos t \; {\bf i} + \sin t\;{\bf j} + 0 \; {\bf k}
\text{ and }
 \cos (t+\tfrac14{\pi})\, \big[{\tfrac32}{\bf i} + 
{ \tfrac  {3\sqrt{3}}2} {\bf k}\big]
+
3\sin(t+\tfrac14{\pi}) \; {\bf j}\;,$$ 
respectively, where $0\le t \le 2\pi\,$.
\begin{questionparts}
\item
Give a geometrical description of the motion of $P$ and $Q$.
\item
Let $\theta$ be the angle $POQ$ at time $t$ that satisfies 
$0\le\theta\le\pi\,$. Show that
\[
\cos\theta = \tfrac{3\surd2}{8} -\tfrac14 \cos( 2t +\tfrac14 \pi)\;.
\]
\item  Show that 
the total time  for which 
$\theta \ge \frac14 \pi$ is $\tfrac32 \pi\,$.
\end{questionparts}

Solution source

\begin{questionparts}
\item $P$ is travelling in a unit circle about the origin in the $\mathbf{i}-\mathbf{j}$ plane. $Q$ is travelling in a circle (also about the origin, but in a different plane with radius $3$).

\item $\,$
\begin{align*}
&& \mathbf{p}\cdot \mathbf{q} &= |\mathbf{p}||\mathbf{q}| \cos \theta \\
\Rightarrow && \cos \theta &= \frac{\tfrac32\cos t \cos(t + \tfrac{\pi}4)+3\sin t \sin (t + \tfrac{\pi}{4})}{3} \\
&&&= \tfrac12\cos t \cos(t + \tfrac{\pi}4)+\sin t \sin (t + \tfrac{\pi}{4}) \\
&&&= \tfrac14 (\cos (2t + \tfrac{\pi}{4}) + \cos(\tfrac{\pi}{4} ))+\tfrac12(\cos(\tfrac{\pi}{4})-\cos(2t + \tfrac{\pi}{4})) \\
&&&= \tfrac{3\sqrt{2}}8 - \tfrac14 \cos ( 2t +\tfrac{\pi}{4})
\end{align*}

\item If $\theta \geq \frac14\pi$, then $\cos \theta \leq \frac{\sqrt{2}}2$

\begin{align*}
&& \frac{\sqrt{2}}2 & \geq  \frac{3\sqrt{2}}8 - \frac14 \cos ( 2t +\tfrac{\pi}{4}) \\
\Rightarrow && \frac{\sqrt{2}}2 &\geq -\cos(2t + \tfrac{\pi}{4}) \\
\Rightarrow && \cos(2t + \tfrac{\pi}{4}) &\geq -\frac{1}{\sqrt{2}} \\
\Rightarrow && 2t + \tfrac{\pi}{4} &\not\in (\tfrac{3\pi}{4},\tfrac{5\pi}{4}) \cup  (\tfrac{11\pi}{4},\tfrac{13\pi}{4}) \\
\Rightarrow && t &\not\in (\tfrac{\pi}{4}, \tfrac{\pi}{2})\cup (\tfrac{5\pi}{4}, \tfrac{3\pi}{2})
\end{align*}

which is is a time of $\frac{\pi}{2}$, therefore the left over time is $\frac32\pi$
\end{questionparts}