The problem requires you to find the position vector $\mathbf{v}$ given the initial point $P = (2, 3)$ and the terminal point $Q = (9, 7)$. To find the position vector:

$\mathbf{v} = Q - P = (x_2 - x_1, y_2 - y_1)$

Step-by-step solution:

1. Subtract the corresponding $x$-coordinates: $x_2 - x_1 = 9 - 2 = 7$

2. Subtract the corresponding $y$-coordinates: $y_2 - y_1 = 7 - 3 = 4$

Thus, the position vector $\mathbf{v}$ is: $\mathbf{v} = (7, 4)$

Writing $\mathbf{v}$ in the form $ai + bj$:

$\mathbf{v} = 7i + 4j$

This is your final answer.

Would you like further clarification or a deeper explanation?

Related Questions:

1. What is a position vector, and why do we subtract the points in this order?
2. How does the formula change if the coordinates of $P$ and $Q$ are reversed?
3. Can you explain how position vectors are used in physics or geometry?
4. How do you convert a position vector into magnitude and direction form?
5. Can you determine the length (magnitude) of this vector? How?

Tip:

Always ensure you subtract the initial point's coordinates from the terminal point's coordinates to find the position vector correctly.