Hyperplane vs Projective Lines

Question

My definition for hyperplanes in $\mathbb{P}^n$ is the locus of pts. $[x_0, \ldots, x_n]$ of $\mathbb{P}^n$ satisfying a linear equation $\sum a_i x_i = 0$ where $(a_0, \ldots, a_n) \neq 0$. So my question is: Aren't hyperplane just equivalent to projective lines? If so why are there two definition for it?

Answer 1

If you forget the projective part, you get all the vectors that are orthogonal to a given nonzero vector $(a_0,\dots, a_n)$.
That is a linear hyperplane ($n$ dimensional subspace) of the $n+1$ dimensional vector space.

Then taking homogeneous coordinates decreases the dimensions by $1$.