The unit vector in the direction of the vector $(7, 24)$ is given by $\left( \frac{7}{25}, \frac{24}{25} \right)$.

To calculate the unit vector corresponding to a given vector, we first need to determine the magnitude (or length) of that vector. The magnitude of a vector represented as $(a, b)$ can be calculated using the formula:

For the vector $(7, 24)$, we substitute the components into the formula:

Once we have the magnitude, we can obtain the unit vector by dividing each component of the original vector by its magnitude. This process normalizes the vector, resulting in a vector of length 1 while preserving its direction. The unit vector $\mathbf{u}$ can be expressed as:

Thus, the unit vector in the direction of $(7, 24)$ is:

This implies that if you visualize the vector $(7, 24)$ and then scale it down so that its length becomes exactly $1$, the resulting vector will have components $\frac{7}{25}$ and $\frac{24}{25}$. This unit vector points in the same direction as the original vector but has a standardized length. It is particularly useful in various applications within mathematics and physics, such as defining directions independently of magnitude.