To express the vector $\overrightarrow{MQ}$ in component form, we need to follow these steps:

Given Information:

• $\overrightarrow{PR} = \begin{pmatrix} 6 \\ -4 \end{pmatrix}$
• $\overrightarrow{RQ} = \begin{pmatrix} -1 \\ 8 \end{pmatrix}$
• $M$ is the midpoint of $\overrightarrow{PR}$.

Steps:

1. Find the coordinates of $M$: Since $M$ is the midpoint of $\overrightarrow{PR}$, we can find $M$ by averaging the coordinates of $P$ and $R$. Given that we don't have the actual coordinates of $P$ and $R$, we can use the vector components of $\overrightarrow{PR}$.

   The midpoint formula for vectors is: $\overrightarrow{M} = \frac{\overrightarrow{P} + \overrightarrow{R}}{2}$

   But since we know the vector $\overrightarrow{PR} = \begin{pmatrix} 6 \\ -4 \end{pmatrix}$, and we assume $P = (0, 0)$ for simplicity (placing $P$ at the origin), then $M$ is simply halfway between $P$ and $R$: $M = \frac{1}{2} \times \overrightarrow{PR} = \frac{1}{2} \times \begin{pmatrix} 6 \\ -4 \end{pmatrix} = \begin{pmatrix} 3 \\ -2 \end{pmatrix}$

2. Find $\overrightarrow{MQ}$: Now we find the vector $\overrightarrow{MQ}$ by subtracting the coordinates of $M$ from $Q$, using the given vector $\overrightarrow{RQ}$. Since $\overrightarrow{RQ} = \begin{pmatrix} -1 \\ 8 \end{pmatrix}$, and $M = (3, -2)$, we can find $\overrightarrow{MQ}$ as: $\overrightarrow{MQ} = \overrightarrow{RQ} - \overrightarrow{RM}$

   We know that $\overrightarrow{RM}$ is equal to $- \overrightarrow{PR} / 2 = \begin{pmatrix} -3 \\ 2 \end{pmatrix}$.

   Now, $\overrightarrow{MQ} = \overrightarrow{RQ} - \overrightarrow{RM} = \begin{pmatrix} -1 \\ 8 \end{pmatrix} - \begin{pmatrix} -3 \\ 2 \end{pmatrix} = \begin{pmatrix} 2 \\ 6 \end{pmatrix}$.

Final Answer:

$\overrightarrow{MQ} = \begin{pmatrix} 2 \\ 6 \end{pmatrix}$

Let me know if you need further details or have any questions! Here are a few more questions to extend the understanding of this concept:

1. How do you calculate the midpoint of a vector using components?
2. What are the properties of midpoints in geometric figures?
3. How do you add or subtract vectors in component form?
4. Can the same method be used to calculate $\overrightarrow{MQ}$ if the coordinates of $P$ and $Q$ were given directly?
5. What is the relationship between the position vectors and geometric transformations in this problem?

Tip: Always remember that finding midpoints in vector form involves averaging the coordinates of the endpoints, making it easier to calculate distances or other related vectors!