Answer: \(\overrightarrow{OP}=\frac25\mathbf a+\frac35\mathbf c\), and \(2\mathbf b=3\mathbf c+2\mathbf a\).

Since \(AP:PC=3:2\), point \(P\) is \(\frac35\) of the way from \(A\) to \(C\).

Now

\(\overrightarrow{AC}=\mathbf c-\mathbf a.\)

Therefore

So

Also \(OP:PB=2:3\), so \(P\) is \(\frac25\) of the way from \(O\) to \(B\). Hence

\(\overrightarrow{OP}=\frac25\mathbf b.\)

Equating the two expressions for \(\overrightarrow{OP}\),

Multiplying by \(5\) gives

\(2\mathbf b=2\mathbf a+3\mathbf c,\)

which is the required result.