2017 NYJC P1 Q9

With reference to an origin \(O\), a particle \(P\) moves in space with position vector \(\left( \lambda -\mu \right)\mathbf{i}+\left( 1+2\mu \right)\mathbf{j}+\left( 2-3\lambda \right)\mathbf{k}\). Another particle \(Q\) moves along the line \(l\) with equation \(\mathbf{r}=\left( \begin{matrix} 1 \\ 1 \\ 0 \\ \end{matrix} \right)+t\left( \begin{matrix} 2 \\ -2 \\ -3 \\ \end{matrix} \right)\), \(t\in \mathbb{R}\).

1. State the locus of \(P\).[1]
2. Determine if the particles \(P\) and \(Q\) can meet.[3]
3. Find the shortest possible distance between \(P\) and \(Q\).[2]

Another particle \(R\) moves along the line \(m\) with equation \(\mathbf{r}=\left( \begin{matrix} 1 \\ k \\ 0 \\ \end{matrix} \right)+s\left( \begin{matrix} 2 \\ -2 \\ -3k \\ \end{matrix} \right)\),\(s\in \mathbb{R}\), where \(k\) is a constant.

4. Find condition(s) satisfied by \(k\) if lines \(l\) and \(m\) are skew lines.[3]
5. A particle is shot from \(X\left( 0,-1,-5 \right)\) perpendicularly toward the path of \(Q\). Find the coordinates of the point where it crosses the path of \(Q\).[2]