Problem source

In 3-dimensional space, the lines $m_1$ and $m_2$ pass through the origin and 
have directions $\bf i + j$ and $\bf i +k $, respectively. Find the directions  of
the two lines $m_3$ and $m_4$  that  pass through the origin and make
angles of $\pi/4$   with both $m_1$ and $m_2$. Find also the cosine of the 
acute angle between $m_3$ and $m_4$.
The points $A$ and $B$ lie on $m_1$ and $m_2$ respectively, and are each at 
distance $\lambda \surd2$ units from~$O$. The points $P$ and $Q$ 
lie on $m_3$ and $m_4$ respectively, and are each at 
distance $1$ unit from~$O$. 
If all the coordinates (with respect to axes $\bf i$, $\bf j$ and $\bf k$)
 of $A$, $B$, $P$ and $Q$ are non-negative, prove that:
\begin{questionparts}
\item there are only two values of $\lambda$ for which $AQ$ is perpendicular
to $BP\,$;
\item there are no non-zero values of $\lambda$ for which $AQ$ and $BP$
intersect.
\end{questionparts}