(a) To show that lines l and m are perpendicular, calculate the dot product of their direction vectors. The direction vector of l is \(4\mathbf{i} - \mathbf{j} + 3\mathbf{k}\) and for m is \(-\mathbf{i} + 2\mathbf{j} + 2\mathbf{k}\).
Calculate the dot product: \((4)(-1) + (-1)(2) + (3)(2) = -4 - 2 + 6 = 0\).
Since the dot product is zero, the lines are perpendicular.
(b) To find the intersection, equate the parametric equations of l and m:
\(3 + 4s = 1 - t\)
\(2 - s = -1 + 2t\)
\(5 + 3s = -2 + 2t\)
Solving these, we find \(s = -1\) and \(t = 2\).
Substitute back to verify all equations are satisfied. The position vector of the intersection is \(-\mathbf{i} + 3\mathbf{j} + 2\mathbf{k}\).
(c) To find the perpendicular distance from the origin to line m, use the formula for the distance from a point to a line in vector form. The point is the origin \(\mathbf{0}\) and the line is \(\mathbf{r} = \mathbf{i} - \mathbf{j} - 2\mathbf{k} + t(-\mathbf{i} + 2\mathbf{j} + 2\mathbf{k})\).
Using the formula, the distance \(d\) is given by:
\(d = \frac{|(-1)(-1) + (-1)(2) + (-2)(2)|}{\sqrt{(-1)^2 + 2^2 + 2^2}} = \frac{|1 - 2 - 4|}{\sqrt{9}} = \frac{5}{3}\).
Thus, the length of the perpendicular is \(\frac{1}{3}\sqrt{5}\).
