Find the length BC to the diagram below. A. |BC| = 3cm C. |BC| = 5cm B. |BC| = 4cm D. |BC| = 6cm

We are given the vectors $\vec{PQ}$ and $\vec{QR}$:

$$\vec{PQ} = \begin{pmatrix} -3 \\ -4 \end{pmatrix}$$ $$\vec{QR} = \begin{pmatrix} 8 \\ -2 \end{pmatrix}$$

The vector $\vec{PR}$ can be found by adding the vectors $\vec{PQ}$ and $\vec{QR}$:

$$\vec{PR} = \vec{PQ} + \vec{QR}$$

$$\vec{PR} = \begin{pmatrix} -3 \\ -4 \end{pmatrix} + \begin{pmatrix} 8 \\ -2 \end{pmatrix}$$

Add the corresponding components:

$$\vec{PR} = \begin{pmatrix} -3 + 8 \\ -4 + (-2) \end{pmatrix}$$ $$\vec{PR} = \begin{pmatrix} 5 \\ -6 \end{pmatrix}$$

The calculated vector $\vec{PR} = \begin{pmatrix} 5 \\ -6 \end{pmatrix}$ matches option B.

The final answer is $\boxed{\text{B. } \vec{PR} = \begin{pmatrix} 5 \\ -6 \end{pmatrix}}$.