To determine the magnitude of the vector $(3, 4)$, we use the formula

The magnitude of a vector provides a measure of its length. For a vector represented by its components $(x, y)$, the magnitude can be calculated using the Pythagorean theorem. This is because the vector forms a right-angled triangle with the x-axis and y-axis, where the components $x$ and $y$ correspond to the lengths of the triangle’s legs.

To find the magnitude of the vector $(3, 4)$, follow these steps:

1. Square the x-component: $$3^2 = 9.$$
2. Square the y-component: $$4^2 = 16.$$
3. Add these squares together: $$9 + 16 = 25.$$
4. Take the square root of the sum: $$\sqrt{25} = 5.$$

Thus, the magnitude of the vector $(3, 4)$ is $5$. This method can be applied to any vector in two-dimensional space, allowing you to understand the distance from the origin $(0, 0)$ in a Cartesian coordinate system.