Norm

Standard Definition: The norm (or magnitude) of a vector is a non-negative value that represents the length or size of the vector in multi-dimensional space. For a vector $\mathbf{v}$ in Euclidean space, its norm is given by:

where $v_1, v_2, \dots, v_n$ are the components of the vector.

ELI5 with Real-life Example: Imagine you're on a treasure hunt. You have a map that tells you to walk 3 steps east and then 4 steps north to find the treasure. Now, instead of walking in two separate directions, you decide to take a shortcut diagonally. The number of steps you'd take on this shortcut is the "norm" or length of your diagonal path. Using the Pythagoras theorem (from right-angled triangles), you'd find that you need to walk 5 steps diagonally to reach the treasure. In this case, the steps east and north are like the components of a vector, and the diagonal path's length (5 steps) is the norm of that vector.

In many real-life scenarios, especially in physics and engineering, the norm helps determine the "strength" or "intensity" of a vector quantity. For example, in electronics, the magnitude of an electric field vector can determine the force felt by charged particles. Similarly, in mechanics, the length of a velocity vector can indicate the speed of an object.