How do you represent the vector from (0, 0) to (9, 12)?

The vector from (0, 0) to (9, 12) is represented as \(\mathbf{v} = \begin{pmatrix} 9 \\ 12 \end{pmatrix}\).

In more detail, a vector is a quantity that has both magnitude and direction. When we talk about the vector from one point to another, we are describing the direction and distance from the starting point to the ending point. In this case, the starting point is (0, 0) and the ending point is (9, 12).

To represent this vector, we look at the change in the x-coordinate and the change in the y-coordinate. The change in the x-coordinate is found by subtracting the x-coordinate of the starting point from the x-coordinate of the ending point: \(9 - 0 = 9\). Similarly, the change in the y-coordinate is \(12 - 0 = 12\).

We then write these changes as a column vector, which is a way of organising the information neatly. The column vector for this example is written as:
\[
\mathbf{v} = \begin{pmatrix} 9 \\ 12 \end{pmatrix}
\]
This tells us that to get from (0, 0) to (9, 12), we move 9 units in the x-direction (right) and 12 units in the y-direction (up).

Vectors are very useful in many areas of mathematics and physics because they allow us to easily describe movements and forces. Understanding how to represent and manipulate vectors is a key skill in GCSE Maths.