Which equation is used to compute the magnitude of a vector?

The equation used to compute the magnitude of a vector is given by the expression that involves the square root of the sum of the squares of its components. This is known as the Pythagorean theorem applied in a two-dimensional space. For a vector represented by its components (x) and (y), the magnitude is calculated as the square root of the sum of the squares of these components.

In mathematical terms, the magnitude (||\mathbf{v}||) of the vector (\mathbf{v} = (x, y)) is expressed as:

[ ||\mathbf{v}|| = \sqrt{x^2 + y^2} ]

This formula holds true because the vector can be visualized as a right triangle where (x) and (y) are the lengths of the legs, and the magnitude is the length of the hypotenuse.

The other options do not correctly represent the calculation of a vector's magnitude. The first choice suggests adding the absolute values of the components, which does not reflect their actual geometric nature. The second option involves the cosine of the components, which relates to angles rather than the lengths of the sides. Lastly, the fourth option presents a subtraction operation