How to find the absolute value of a vector?

Keep in mind that absolute value is distance from zero. So you can use the distance formula to find the absolute value:

$$ \sqrt{x^2+y^2+z^2} $$

Absolute value of a vector means taking second norm of the vector i.e. $\|x\|$. That means the same thing as $\sqrt{x_1^2 +x_2^2+...+x_n^2}$. I don't understand why some top researchers in computer science abuse the notation where $|x|$ is widely used for absolute value of scalars in math.

I have also seen papers using $|x|$ to mean absolute value of each component of the vector $x$. In summary, meaning will depends upon the context and author.