Given PQ = 2 3 and OQ = -1 4 . Find the coordinates of P.

The vector $\vec{PQ}$ can be expressed as the position vector of Q minus the position vector of P:

$$\vec{PQ} = \vec{OQ} - \vec{OP}$$

We want to find the coordinates of P, which are given by the position vector $\vec{OP}$. Rearrange the formula to solve for $\vec{OP}$:

$$\vec{OP} = \vec{OQ} - \vec{PQ}$$

Subtract the corresponding components:

$$\vec{OP} = \begin{pmatrix} -1 - 2 \\ 4 - 3 \end{pmatrix}$$ $$\vec{OP} = \begin{pmatrix} -3 \\ 1 \end{pmatrix}$$

The coordinates of P are $\boxed{(-3, 1)}$.

4) The vectors $a = \begin{pmatrix} 2 \\ 5 \end{pmatrix}$, $b = \begin{pmatrix} m \\ -3 \end{pmatrix}$ and $c = \begin{pmatrix} 8 \\ n \end{pmatrix}$ are such that $a + 2b = c$. Find the values of m and n.

$$\begin{pmatrix} 2 \\ 5 \end{pmatrix} + \begin{pmatrix} 2m \\ -6 \end{pmatrix} = \begin{pmatrix} 8 \\ n \end{pmatrix}$$ $$\begin{pmatrix} 2 + 2m \\ 5 + (-6) \end{pmatrix} = \begin{pmatrix} 8 \\ n \end{pmatrix}$$ $$\begin{pmatrix} 2 + 2m \\ -1 \end{pmatrix} = \begin{pmatrix} 8 \\ n \end{pmatrix}$$

For the x-components:

$$2 + 2m = 8$$

For the y-components:

$$-1 = n$$

From the y-components, we directly get:

$$n = -1$$

From the x-components:

$$2m = 8 - 2$$ $$2m = 6$$ $$m = \frac{6}{2}$$ $$m = 3$$

The values are $\boxed{m = 3, n = -1}$.