The magnitude of the vector $(1, 1)$ is given by $\sqrt{2}$.

To determine the magnitude of a vector, we can apply the Pythagorean theorem. A two-dimensional vector, such as $(1, 1)$, can be visualized as the hypotenuse of a right-angled triangle, where the components of the vector represent the lengths of the two perpendicular sides. In this case, both sides measure $1$ unit in length.

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

For the vector $(1, 1)$, we substitute $a = 1$ and $b = 1$ into the formula, resulting in

This simplifies to

Thus, the magnitude of the vector $(1, 1)$ is indeed $\sqrt{2}$.

Understanding the magnitude of a vector is crucial, as it indicates the length or size of the vector, independent of its direction. This concept is widely applicable in fields such as physics, engineering, and computer graphics. For GCSE Maths students, mastering the calculation of a vector’s magnitude is a foundational skill that aids in the resolution of more complex vector-related problems.