Why is this the expression of the line between two points in the projective space?

Question

Let $P,Q\in \mathbb{P}^3$ be two points in the projective space. I have read that the line of $\mathbb{P}^3$ that goes though these two points is the set of points of the form $\{Ps+Qt \: : \: [s,t]\in \mathbb{P}^1\}$. I don't really get why this is like this, and would very much appreciate if someone could help me get an intuition of this. In class we've defined the lines of the projective space as the common zero locus of $2$ homogeneous independent equations and I don't rellay see how this expression gives me two such equations.

Answer 1

In a non-projective setup you might write the points of a line e.g. as

$$P+t(Q-P)$$

and read that as point on line plus any multiple of the direction of the line. You can also turn this into

$$(1-t)P+tQ$$

by simple rearrangement and then to

$$sP+tQ\qquad\text{where }s+t=1$$

by introducing an additional name.

For homogeneous coordinates, multiples of a vector describe the same point. So you can weaken the $s+t=1$ condition to $(s,t)\neq(0,0)$, i.e. they must not be zero at the same time. This gives you the same line, but also includes the point at infinity which would be $s=-t$ if both vectors have a $1$ in their last entry.