$$\vec{AC}=-\mathbf{a}+\mathbf{c}$$
$$\vec{MQ}=\frac{1}{2}\mathbf{a}+\mathbf{c}-\frac{1}{4}\mathbf{a}$$
$$=\frac{1}{4}\mathbf{a}+\mathbf{c}$$
$$\vec{OP}=\vec{OM}+\vec{MP}$$
$$=\vec{OA}+\vec{AP}$$
$$\vec{MP}=\lambda(\frac{1}{4}\mathbf{a}+\mathbf{c})$$
$$=\frac{\lambda}{4}\mathbf{a}+\lambda \mathbf{c}$$
$$\vec{AP}=\mu(-\mathbf{a}+\mathbf{c})$$
$$=-\mu\mathbf{a}+\mu\mathbf{c}$$
As $$\vec{AP}$$ and $$\vec{MP}$$ intersect, if I let them equal, I can find the $$P$$.
$$\frac{\lambda}{4}\mathbf{a}+\lambda \mathbf{c}$$ $$=-\mu\mathbf{a}+\mu\mathbf{c}$$
Therefore, $$\frac{\lambda}{4}=-\mu$$ and $$\lambda = \mu$$
Then if I subtitute $$\mu$$ to $$\frac{\lambda}{4}=-\mu$$,
$$\frac{\mu}{4}=-\mu$$
$$\mu=-4$$
Sub -4 to $$\vec{MP}=\lambda(\frac{1}{4}\mathbf{a}+\mathbf{c})$$ (As $$\mu = \lambda$$)
$$\vec{MP}=-4(\frac{1}{4}\mathbf{a}+\mathbf{c})$$
$$=-\mathbf{a}-4\mathbf{c}$$
Finally,
$$\vec{OP} = \vec{OM}+\vec{MP}$$
$$=\frac{1}{2}\mathbf{a}+(-\mathbf{a}-4\mathbf{c})$$
$$=\frac{1}{2}\mathbf{a}-\mathbf{a}-4\mathbf{c}$$
$$=-\frac{1}{2}\mathbf{a}-4\mathbf{c}$$
