Show that if $\Gamma \subseteq \mathbb{P}^n$ consists of d points and is not contained in a line, then $\Gamma$ may be described as the zero locus of polynomials of degree $d - 1$ and less.
This problem is exercise 1.3 from Joe Harris's Algebraic Geometry: A First Course. I don't so much want an answer to the original question as some help understanding it. My issue is a general unfamiliarity with projective geometry, and I think some concrete examples might help me better make sense of the situation.
More specifically, just above this problem the book states that for a line $L \subset \mathbb{P}^n$,
...it's not hard to see that a polynomial $F(Z)$ of degree $d - 1$ or less that vanishes on $d$ points $p_i \in L$ will vanish identically on L.
But I'm not sure what it means in any useful, algebraic sense for points in projective space to be collinear. I can't even seem to put together for my own use three collinear points in $\mathbb{P^3}$ to see why (in a simple numeric sense) every quadratic equation that is zero at all three of them will also be zero everywhere else on the line.