(a) To find the values of \(a\), \(b\), and \(c\), we use the fact that lines l and m are perpendicular. The direction vectors of l and m are \((c, -2, 4)\) and \((2, -3, 1)\) respectively. The scalar product of these vectors must be zero:

\(2c + 6 + 4 = 0\)

Solving gives \(c = -5\).

Since P lies on l, substitute \(\mathbf{r} = 4\mathbf{i} + 7\mathbf{j} - 2\mathbf{k}\) into the equation of l:

\(4 = a + \lambda c\)

\(7 = 3 - 2\lambda\)

\(-2 = b + 4\lambda\)

Solving these equations gives \(a = -6\), \(b = 6\), and \(\lambda = 2\).

(b) To find the position vector of R, first find the position vector of Q by setting the perpendicular from P to m:

\(\mathbf{OQ} = -\mathbf{i} + 5\mathbf{j} + 2\mathbf{k}\)

Using the ratio \(PQ : QR = 2 : 3\), find \(\mathbf{OR}\):

\(\mathbf{OR} = \mathbf{OP} + \frac{5}{2} \mathbf{PQ}\)

\(\mathbf{OR} = \left(4, 7, -2\right) + \frac{5}{2} \left(-5, -2, 4\right)\)

\(\mathbf{OR} = \left(\frac{17}{2}, \frac{1}{2}, 8\right)\)