How do you determine the magnitude of vector (14, -48)?

To determine the magnitude of vector (14, -48), use the Pythagorean theorem: √(14² + (-48)²).

In more detail, the magnitude of a vector is essentially its length. For a vector given by its components (x, y), the magnitude can be found using the Pythagorean theorem. This is because the vector forms a right-angled triangle with the x and y axes. The magnitude is the hypotenuse of this triangle.

For the vector (14, -48), the x-component is 14 and the y-component is -48. According to the Pythagorean theorem, the magnitude (or length) of the vector is the square root of the sum of the squares of its components. Mathematically, this is written as:

\[ \text{Magnitude} = \sqrt{x^2 + y^2} \]

Substituting the given values:

\[ \text{Magnitude} = \sqrt{14^2 + (-48)^2} \]

First, calculate the squares of the components:

\[ 14^2 = 196 \]
\[ (-48)^2 = 2304 \]

Next, add these squares together:

\[ 196 + 2304 = 2500 \]

Finally, take the square root of the sum:

\[ \sqrt{2500} = 50 \]

So, the magnitude of the vector (14, -48) is 50. This method can be applied to any vector to find its magnitude, making it a useful tool in many areas of mathematics and physics.