Definition 11.2 in Joe Harris's Algebraic Geometry: A First Course is the following.
The dimension of an irreducible quasi-projective variety $X\subseteq P^n$ is the smallest integer $k$ such that a general $(n-k-1)$-plane $\Lambda\subseteq P^n$ is disjoint from $X$.
I am unable to follow what the author means by a "general" $(n-k-1)$-plane (apart from the vague intuitive meaning). I tried to look at the index but there is no mention of the word "general". Can somebody throw some light on what is the definition trying to say?
Planes of codimension $k+1$ are parametrized by $G(n-k,n+1)$. So the statement means if there is a non-empty Zariski open $U$ so that for all $\Lambda \in U$, $\Lambda$ is disjoint from $X$.