The vector from the point $(0, 0)$ to the point $(9, 12)$ is denoted as

To elaborate, a vector is a mathematical object that possesses both magnitude and direction. When we refer to the vector connecting two points, we are describing both the distance and the direction from the initial point to the final point. Here, the initial point is $(0, 0)$, and the terminal point is $(9, 12)$.

To represent this vector, we calculate the changes in the x-coordinate and y-coordinate. The change in the x-coordinate is determined by subtracting the x-coordinate of the starting point from that of the ending point:

Similarly, the change in the y-coordinate is calculated as

We can then express these changes in a structured format known as a column vector. For this example, the column vector is represented as:

This representation indicates that to move from $(0, 0)$ to $(9, 12)$, one must move $9$ units in the x-direction (to the right) and $12$ units in the y-direction (upward).

Vectors are invaluable in various fields of mathematics and physics, as they provide a straightforward way to describe movements and forces. Mastering the representation and manipulation of vectors is an essential skill in GCSE Maths.