(a) \(M=(3,1,0)\), and \(N=A+\tfrac23(B-A)=(10/3,2,2)\). Hence \[\boxed{\mathbf r=3\mathbf i+\mathbf j+\lambda(\tfrac13\mathbf i+\mathbf j+2\mathbf k)}.\]

(b) On OB, \(\mathbf r=\mu(2,2,3)\). Equating coordinates to \((3+\lambda/3,1+\lambda,2\lambda)\) gives \(\lambda=3,\mu=2\), so \(\boxed{\overrightarrow{OP}=4\mathbf i+4\mathbf j+6\mathbf k}\).

(c) Use vectors from the vertex P: \(\overrightarrow{PO}=(-4,-4,-6)\), \(\overrightarrow{PM}=(-1,-3,-6)\). Their dot product is 52, so \[\cos\angle OPM=\frac{52}{\sqrt{68}\sqrt{46}},\qquad\boxed{\angle OPM\approx21.6^\circ}.\]