The points \(A\) and \(B\) have position vectors
\(\bf i +j+k\)
and \(5{\bf i} - {\bf j} -{\bf k}\), respectively, relative to the origin \(O\).
Find \(\cos2\alpha\), where \(2\alpha\) is the angle \(\angle AOB\).
The line \(L _1\) has equation
\({\bf r} =\lambda(m{\bf i}+n {\bf j} + p{\bf k})\).
Given that \(L _1\) is inclined equally to \(OA\) and to \(OB\),
determine a relationship between \(m\), \(n\) and~\(p\).
Find
also values of \(m\), \(n\) and~\(p\) for which \(L _1\) is the
angle bisector of \(\angle AOB\).
The line \(L _2\) has equation
\({\bf r} =\mu(u{\bf i}+v {\bf j} + w{\bf k})\).
Given that \( L _2\) is inclined at an angle \(\alpha\) to \(OA\),
where \(2\alpha = \angle AOB\), determine a relationship between
\(u\), \(v\) and \(w\).
Hence describe the surface with Cartesian equation
\(x^2+y^2+z^2 =2(yz+zx+xy)\).