N2000 P2 Q15 [Modified]

Relative to the origin \(O\), the points \(A\), \(B\), \(C\) have position vectors \(5\mathbf{i}+4\mathbf{j}+10\mathbf{k}\), \(-4\mathbf{i}+4\mathbf{j}-2\mathbf{k}\) and \(-5\mathbf{i}+9\mathbf{j}+5\mathbf{k}\) respectively.

1. Find the cartesian equation of the line \(AB\).[2]
2. Find the length of projection of the vector \(\overrightarrow{AC}\) on line \(AB\).[2]
3. Find the position vector of the foot of the perpendicular,\(N\), from \(C\) to line \(AB\).[3]
4. The point \(D\) lies on the line \(CN\) produced and is such that \(N\) is the mid-point of \(CD\).
   Find the position vector of \(D\).[2]