# show that $(𝑎, 𝑏, 𝑐)$ and $(𝑎′, 𝑏′, 𝑐′)$ define the same projective line if $(𝑎, 𝑏, 𝑐) = 𝜆(𝑎′, 𝑏′, 𝑐′)$ for some nonzero $𝜆 ∈ ℂ$ [closed]

How do I show that the triples $$(𝑎, 𝑏, 𝑐)$$ and $$(𝑎′, 𝑏′, 𝑐′)$$ define the same projective line if and only if $$(𝑎, 𝑏, 𝑐) = 𝜆(𝑎′, 𝑏′, 𝑐′)$$ for some nonzero $$𝜆 ∈ ℂ$$

I've seen that this is the definition of a projective plane, but I don't know how to actyally prove this. Any help would be great!!

Edit: A projective line in $$ℙ^2(ℂ)$$ is defined by $$𝑎 𝑥 + 𝑏 𝑦 + 𝑐 𝑧 = 0$$ where $$0 ≠ (𝑎, 𝑏, 𝑐) ∈ ℂ^3$$

• What definition of $\mathbb{C}P^2$ are you using? Usually this is just the definition of the defining equivalence relation on $\mathbb{C}^3$. – Nate Gallup Apr 8 at 2:47
• I added the definition in the post! – user65432109 Apr 8 at 2:58
• Projective lines are just equivalence classes of hyperplanes in $\mathbb{C}^3$. – mathmathmath Apr 8 at 3:00

This is straightforward. You show that any point $$(x,y,z)$$ on the projective line $$(a,b,c)$$ is also on the projective line $$(a’,b’,c’)$$; and you also show the converse i.e. that any point $$(x,y,z)$$ on line $$(a’,b’,c’)$$ is also on line $$(a,b,c)$$.