# Can a component of velocity be greater than the speed of light?

Suppose a particle is travelling with a velocity v. Can a component of this velocity(let's say the y-component) be greater than the speed of light c?

• no, because then the total speed would be greater than c.
– Bob
Jun 30, 2017 at 13:49
• It is trivial that since the norm of the velocity can't be greater than the speed of light, none of its component can be. Do you have any other confusion or a paradox that leads you to investigate this question?
– user87745
Jun 30, 2017 at 13:57
• Well, I was studying superluminal motion from Wikipedia. There is a concept where the apparent velocity was greater than the speed of light. I got confused between that and component of a velocity. Here's the link: - en.wikipedia.org/wiki/Superluminal_motion Jun 30, 2017 at 15:29

No, because the total magnitude of the speed vector may not exceed the speed of light, and it is itself greater than the absolute value any of its component :

$$\vec{v}=\begin{bmatrix}x\\y\\z\end{bmatrix}$$

Since:

$|x|=\sqrt{x^2}\le\sqrt{x^2+y^2+z^2}=||\vec{v}||$

Physically, the norm may not exceed the speed of light $c$: $$||\vec{v}||<c$$

Therefore $|x|<c$, $|y|<c$ and $|z|<c$

• Well, I was studying superluminal motion from Wikipedia. There is a concept where the apparent velocity was greater than the speed of light. I got confused between that and component of a velocity. Here's the link: - en.wikipedia.org/wiki/Superluminal_motion It would be helpful if we can clear this doubt. Jun 30, 2017 at 15:36
• @QuamosM87 Those are indeed distinct questions. It looks like a good Stack Overflow question, as well. Why don't you post a new one/search for an existing one? Jul 5, 2017 at 11:47