I have the following definitions:

Given a vector space V over a field k, we can define the projective space PV=(V∖{0})/∼ where ∼ identifies all points that lie on the same line through the origin.

A projective subspace PW of PV is of the form π(W∖{0}), where π is the residue class map and W is a vector subspace of V. Define dim(PV)=dim(V)−1. A line in PV is a 1-dimensional projective subspace.

Now I’m finding it difficult to visualise what a line in projective space actually is. I can understand why any two lines in a projective plane intersect. Suppose I’m in P3 and want to write ‘an equation’ for the line that goes through the points p=(1:0:0:0) and q=(a:b:c:d). How could I do that? Does my question even make sense? I’m concerned because PV isn’t actually a vector space, so can I think of points inside it as vectors?

Thanks

**Answer**

If you have two distinct points A=[a0:…:an],B=[b0:…:bn]∈Pn, they correspond to two vectors a=(a0,…,an),b=(b0,…,bn)∈kn+1.

These vectors span a plane Λ⊂kn+1 whose vectors are the ua+vb,(u,v∈k).

The corresponding line ¯AB=P(Λ)⊂Pn has its points of the form [ua0+vb0:…:uan+vbn](u,v∈k, not both zero ).

In your particular case the projective line ¯pq joining p=[1:0:0:0] and q=[a:b:c:d] has points with coordinates [u+va:vb:vc:vd]

**Attribution***Source : Link , Question Author : Jonathan , Answer Author : Georges Elencwajg*