$|x|$ symbol for x in $R^3$

Question

I'm doing my assignment which has a notation like this: $P_1={(x,y,z)\ \in\ \mathbb{R}^3:\ |x|\ \le1,\ |y|\ \le1,\ |z|\ \le1}$. At first I thought $|x|\ =\ \sqrt{{x_1}^2+{x_2}^2+{x_3}^2}$, which is the magnitude of 3D vector x. When I asked my professor he said |x| is the absolute value of x. But I read in many sources saying there's no absolute value for a vector, there's only $||x||\ =\ \sqrt{{x_1}^2+{x_2}^2+{x_3}^2}$. So I want to ask if what |x| actually is and how its formula looks like? Thank you in advance!

Answer 1

Since $P_1\in\Bbb R^3$ and $P_1=(x,y,z)$, $x$, $y$, and $z$ are real numbers, rather than elements of $\Bbb R^3$. So, saying that $x$ is just the absolute value of $x$ makes sense.