Answer: \(\overrightarrow{OP}=\lambda(\mathbf a+\mathbf c)\), \(\overrightarrow{OP}=\mathbf a+\mu\left(-\mathbf a+\dfrac25\mathbf c\right)\), and \(DP:PA=OP:PB=2:5\).

We start with the main method. Use the measurements and relationships shown in the diagram, then translate them into algebraic or trigonometric equations. Use position vectors and ratios along a line, then compare coefficients of independent vectors.

In the parallelogram,

\(\overrightarrow{OB}=\mathbf a+\mathbf c.\)

Since \(\overrightarrow{OP}=\lambda\overrightarrow{OB}\),

\(\overrightarrow{OP}=\lambda(\mathbf a+\mathbf c).\)

Also, \(D\) divides \(OC\) in the ratio \(2:3\), so

\(\overrightarrow{OD}=\frac25\mathbf c.\)

Hence

\(\overrightarrow{AD}=\overrightarrow{OD}-\overrightarrow{OA}=\frac25\mathbf c-\mathbf a.\)

Since \(\overrightarrow{AP}=\mu\overrightarrow{AD}\),

\(\overrightarrow{OP}=\overrightarrow{OA}+\overrightarrow{AP} =\mathbf a+\mu\left(-\mathbf a+\frac25\mathbf c\right).\)

So the two expressions for \(\overrightarrow{OP}\) are

\(\lambda(\mathbf a+\mathbf c)\)

and

\(\mathbf a+\mu\left(-\mathbf a+\frac25\mathbf c\right).\)

Equate coefficients of \(\mathbf a\) and \(\mathbf c\):

\(\lambda=1-\mu,\qquad \lambda=\frac25\mu.\)

Thus

\(1-\mu=\frac25\mu.\)

So \(5-5\mu=2\mu\), giving

\(\mu=\frac57.\)

Then

\(\lambda=\frac25\cdot\frac57=\frac27.\)

Since \(\lambda=\frac27\), point \(P\) divides \(OB\) in the ratio

\(OP:PB=2:5.\)

Since \(\mu=\frac57\), \(AP:PD=5:2\), and therefore

\(DP:PA=2:5.\)

Hence \(P\) divides both \(DA\) and \(OB\) in the ratio \(2:5\).

This completes the solution and gives the required result.