Question

Given a vector u, the familiar absolute-value notation |u| is often used for its magnitude.
Thus the expressions u•u and |u|2 both mean the same thing. What exactly do they mean?

Answer 1

let u = [a,b]
then u•u = a^2 + b^2

also |u| = √(a^2 + b^2)
then |u|^2 = a^2 + b^2

A geometric interpretation would be the area of a square whose sides equals the magnitude of the vector.

e.g. let u = [3,4]
then |u| = √(9+16) = 5
so the magnitude of the vector is 5
a square with sides 5 would have an area of 25
and |u|^2 = 25

Don't know if this is what you are after.