The magnitude of the vector $(15, 20)$ is $25$.

To determine the magnitude of a vector, we apply the Pythagorean theorem. The vector $(15, 20)$ can be represented as the two perpendicular sides of a right-angled triangle, where $15$ and $20$ are the lengths of these sides. The magnitude of the vector corresponds to the length of the hypotenuse of this triangle.

The formula for calculating the magnitude of a vector $(a, b)$ is given by:

In this scenario, we have $a = 15$ and $b = 20$. Substituting these values into the formula yields:

Thus, the magnitude of the vector $(15, 20)$ is $25$. This indicates that if you were to graph this vector, the straight-line distance from the origin $(0, 0)$ to the point $(15, 20)$ would be $25$ units. Understanding vector magnitude is essential in various fields, including physics, engineering, and computer graphics, where the size and direction of vectors play a critical role.