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So I have a vector which I need to show is timelike. The vector is

[itex] v^{\mu}=t^{\mu}-\frac{t^{\mu}\cdot X^{\mu\prime}}{X^{\mu\prime}\cdot X^{\mu\prime}}X^{\mu\prime} [/itex],

where [itex]t^{\mu}[/itex] is a timelike vector and [itex]X^{\mu\prime}[/itex] is spacelike, however these two vectors are not perpendicular so their dot product does not vanish.

I understand that in order to show that [itex]v^{\mu}[/itex] is timelike, I need to find [itex]v_{\mu}v^{\mu}[/itex] and show that this is greater than 0. So:

[itex]v_{\mu}v^{\mu}=t^{2}-2\frac{(t\cdot X')^{2}}{X^{\prime}\cdot X^{\prime}}+(t\cdot X^{\prime})^{2}[/itex]

So first of all I dont know if this is correct (yes I dont know how to take a dot product obviously lol) and secondly, even if it is, how do I show that this is greater than 0?

Thanks guys!