Problems

1. The line \(L_1\) has vector equation $\displaystyle {\bf r} = \begin{pmatrix} 1 \\ 0 \\ 2 \end{pmatrix} + \lambda \begin{pmatrix} \hphantom{-} 2 \\ \hphantom{-} 2 \\ -3 \end{pmatrix} $. The line \(L_2\) has vector equation $\displaystyle {\bf r} = \begin{pmatrix} \hphantom{-} 4 \\ -2 \\ \hphantom{-} 9 \end{pmatrix} + \mu \begin{pmatrix} \hphantom{-} 1 \\ \hphantom{-} 2 \\ -2 \end{pmatrix} . $ Show that the distance \(D\) between a point on \(L_1\) and a point on \(L_2\) can be expressed in the form \[ D^2 = \left(3\mu -4 \lambda-5 \right)^2 + \left( \lambda -1 \right)^2 + 36\,. \] Hence determine the minimum distance between these two lines and find the coordinates of the points on the two lines that are the minimum distance apart.
2. The line \(L_3\) has vector equation ${\bf r} = \begin{pmatrix} 2 \\ 3 \\ 5 \end{pmatrix} + \alpha \begin{pmatrix} 0 \\ 1 \\ 0 \end{pmatrix} . $ The line \(L_4\) has vector equation $ {\bf r} = \begin{pmatrix} \hphantom{-} 3 \\ \hphantom{-} 3 \\ -2 \end{pmatrix} + \beta \begin{pmatrix} \, 4k\\ 1-k \\ \!\!\! -3k \end{pmatrix} . $ Determine the minimum distance between these two lines, explaining geometrically the two different cases that arise according to the value of \(k\).

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:

1. there are only two values of \(\lambda\) for which \(AQ\) is perpendicular to \(BP\,\);
2. there are no non-zero values of \(\lambda\) for which \(AQ\) and \(BP\) intersect.

Two non-parallel lines in 3-dimensional space are given by \(\mathbf{r}=\mathbf{p}_{1}+t_{1}\mathbf{m}_{1}\) and \(\mathbf{r}=\mathbf{p}_{2}+t_{2}\mathbf{m}_{2}\) respectively, where \(\mathbf{m}_{1}\) and \(\mathbf{m}_{2}\) are unit vectors. Explain by means of a sketch why the shortest distance between the two lines is \[ \frac{\left|(\mathbf{p}_{1}-\mathbf{p}_{2})\cdot(\mathbf{m}_{1}\times\mathbf{m}_{2})\right|}{\left|(\mathbf{m}_{1}\times\mathbf{m}_{2})\right|}. \]

1. Find the shortest distance between the lines in the case \[ \mathbf{p}_{1}=(2,1,-1)\qquad\mathbf{p}_{2}=(1,0,-2)\qquad\mathbf{m}_{1}=\tfrac{1}{5}(4,3,0)\qquad\mathbf{m}_{2}=\tfrac{1}{\sqrt{10}}(0,-3,1). \]
2. Two aircraft, \(A_{1}\) and \(A_{2},\) are flying in the directions given by the unit vectors \(\mathbf{m}_{1}\) and \(\mathbf{m}_{2}\) at constant speeds \(v_{1}\) and \(v_{2}.\) At time \(t=0\) they pass the points \(\mathbf{p}_{1}\) and \(\mathbf{p}_{2}\), respectively. If \(d\) is the shortest distance between the two aircraft during the flight, show that \[ d^{2}=\frac{\left|\mathbf{p}_{1}-\mathbf{p}_{2}\right|^{2}\left|v_{1}\mathbf{m}_{1}-v_{2}\mathbf{m}_{2}\right|^{2}-[(\mathbf{p}_{1}-\mathbf{p}_{2})\cdot(v_{1}\mathbf{m}_{1}-v_{2}\mathbf{m}_{2})]^{2}}{\left|v_{1}\mathbf{m}_{1}-v_{2}\mathbf{m}_{2}\right|^{2}}. \]
3. Suppose that \(v_{1}\) is fixed. The pilot of \(A_{2}\) has chosen \(v_{2}\) so that \(A_{2}\) comes as close as possible to \(A_{1}.\) How close is that, if \(\mathbf{p}_{1},\mathbf{p}_{2},\mathbf{m}_{1}\) and \(\mathbf{m}_{2}\) are as in (i)?

The distinct points \(O\,(0,0,0),\) \(A\,(a^{3},a^{2},a),\) \(B\,(b^{3},b^{2},b)\) and \(C\,(c^{3},c^{2},c)\) lie in 3-dimensional space.

1. Prove that the lines \(OA\) and \(BC\) do not intersect.
2. Given that \(a\) and \(b\) can vary with \(ab=1,\) show that \(\angle AOB\) can take any value in the interval \(0<\angle AOB<\frac{1}{2}\pi\), but no others.