The vector representation of the movement from the point $(4, 2)$ to the point $(7, 8)$ is given by the vector $(3, 6)$.

To clarify this concept, let’s break it down step by step. A vector fundamentally describes both a direction and a magnitude (or distance) from one point to another. In our case, we are transitioning from the point $(4, 2)$ to the point $(7, 8)$.

To determine the vector, we need to compute the differences in the x-coordinates and y-coordinates of these two points. The x-coordinate of the starting point is $4$, while the x-coordinate of the ending point is $7$. The difference in the x-coordinates is calculated as:

Similarly, the y-coordinate of the starting point is $2$, and the y-coordinate of the ending point is $8$. The difference in the y-coordinates is:

Consequently, the vector that represents the transition from $(4, 2)$ to $(7, 8)$ is $(3, 6)$. This indicates a movement of $3$ units to the right (along the x-axis) and $6$ units upward (along the y-axis). In vector notation, this is typically expressed as $(x, y)$, where $x$ corresponds to the horizontal change and $y$ corresponds to the vertical change.

Understanding vectors is essential in various fields of mathematics and physics, as they effectively describe quantities that possess both direction and magnitude. In this illustration, the vector $(3, 6)$ concisely encapsulates the movement from one point to another in a two-dimensional space.