The magnitude of the vector $(7, 24)$ is $25$.

To determine the magnitude of a vector, we apply the Pythagorean theorem. In a two-dimensional space, a vector can be represented as $(x, y)$, where $x$ and $y$ denote its components along the x-axis and y-axis, respectively. The magnitude of the vector represents the length of the line segment that extends from the origin $(0, 0)$ to the point $(x, y)$.

For the vector $(7, 24)$, we can visualize it as forming a right-angled triangle with the x-axis and y-axis. Here, the x-component is $7$, and the y-component is $24$. According to the Pythagorean theorem, the magnitude of the vector corresponds to the hypotenuse of this right triangle.

The formula for calculating the magnitude is given by:

Substituting the values for $x$ and $y$ into the formula yields:

Calculating further:

This simplifies to:

Thus, we find that:

Consequently, the magnitude of the vector $(7, 24)$ is $25$. This implies that if you were to graph this vector, the distance from the origin to the point $(7, 24)$ would measure $25$ units. Understanding the concept of vector magnitude is essential in vector mathematics and has significant applications in fields such as physics and engineering, where it is used to ascertain distances and lengths.