Vectors are commonly used to model forces such as wind, sea current, gravity, and electromagnetism. Calculating the magnitude of vectors is essential for all sorts of problems where forces collide.

Magnitude is defined as the length of a vector. The notation for absolute value

is also used for the magnitude of a vector. For example, |v| refers to magnitude of the vector v. (By the way, some textbooks represent magnitude with double bars

instead of single bars. Either way, the meaning is the same.)

You calculate the magnitude of a vector

by using a variation of the distance formula. This formula is itself a variation of the trusty Pythagorean theorem:

For example, you can calculate the magnitude of the vector

as follows:

As you can see, the use of the absolute value bars for the magnitude of vectors is appropriate: Magnitude, like all other distances, is always measured as a nonnegative value. The magnitude of a vector is the distance from the origin of a graph to its tip, just as the absolute value of a number is the distance from 0 on a number line to that number.