To determine the magnitude of the vector $(3, 4)$, we can apply the Pythagorean theorem, which states that the magnitude can be calculated as follows:

Let’s delve into this concept further. The magnitude of a vector represents its length in space. For a vector expressed in terms of its components $(x, y)$, the magnitude can be computed using the Pythagorean theorem. This is because the vector creates a right-angled triangle with the $x$ and $y$ axes, where the components $(3, 4)$ correspond to the lengths of the two shorter sides of the triangle.

To calculate the magnitude, follow these steps: first, square each component of the vector. For our vector $(3, 4)$, we compute:

• $3^2 = 9$
• $4^2 = 16$

Next, we sum these squares:

Finally, we take the square root of the total:

Thus, the magnitude of the vector $(3, 4)$ is $5$.

This method is applicable to any vector in two-dimensional space and is a fundamental principle in vector mathematics. Mastering the calculation of a vector’s magnitude is essential in various fields of mathematics and physics, as it aids in understanding the size and direction of quantities such as force and velocity.