What is the notation for a vector from (2, 3) to (5, 7)?

The notation for a vector from (2, 3) to (5, 7) is \(\begin{pmatrix} 3 \\ 4 \end{pmatrix}\).

To find the vector from one point to another, you subtract the coordinates of the starting point from the coordinates of the ending point. In this case, the starting point is (2, 3) and the ending point is (5, 7).

First, subtract the x-coordinates: \(5 - 2 = 3\). This gives you the x-component of the vector. Next, subtract the y-coordinates: \(7 - 3 = 4\). This gives you the y-component of the vector.

So, the vector from (2, 3) to (5, 7) is \(\begin{pmatrix} 3 \\ 4 \end{pmatrix}\). This notation means that to get from the point (2, 3) to the point (5, 7), you move 3 units to the right (along the x-axis) and 4 units up (along the y-axis).

Vectors are often written in column form like this to clearly show the direction and magnitude of the movement. The top number represents the horizontal change, and the bottom number represents the vertical change. This method is very useful in many areas of mathematics and physics, as it helps to visualise and calculate movements and forces.