What does "magnitude of the vector" mean?

Explanation:

The magnitude of a vector is the length of the vector. The magnitude of the vector a is denoted as ∥a∥. See the introduction to vectors for more about the magnitude of a vector.

Formulas for the magnitude of vectors in two and three dimensions in terms of their coordinates are derived in this page. For a two-dimensional vector a=(a1,a2), the formula for its magnitude is

∥a∥=a21+a22‾‾‾‾‾‾‾√.

For a three-dimensional vector a=(a1,a2,a3), the formula for its magnitude is

∥a∥=a21+a22+a23‾‾‾‾‾‾‾‾‾‾‾‾√.

The formula for the magnitude of a vector can be generalized to arbitrary dimensions. For example, if a=(a1,a2,a3,a4) is a four-dimensional vector, the formula for its magnitude is

∥a∥=a21+a22+a23+a24‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾√.

conservation of mass

mass before something happens = mass afterwards