What is $|x-y|$ here?

Question

I feel confused about this paragraph. What does $|x-y|$ mean here? Taking every element of a vector positive? Or length of a vector? I feel so confused. Thanks!

Answer 1

It's the length of the vector $x-y$. You can see this in the line that follows: $$ x-y=\sqrt{(x-y)\cdot(x-y)}=\sqrt{x\cdot x+y\cdot y-2x\cdot y}=\sqrt{|x|^2+|y|^2-2x\cdot y}. $$