I am going slowly through this book "Vector Calculus"
by Peter Baxandall (Author), Hans Liebeck (Author).
https://www.amazon.com/Vector-Calculus-Dover-Books-Mathematics/dp/0486466205
On page 69-70 there's this definition (see below), and I am confused about point [iv] from it.
The book does have provide a prior definition of what a plane means in $\mathbb{R}^n$.
I thought a plane in $\mathbb{R}^n$ always means a hyperplane i.e. an object of $n-1$ dimensions determined by a point $A$ in $\mathbb{R}^n$ and by $(n-1)$ linearly independent vectors (e.g. $(n-1)$ pairwise orthogonal non-zero vectors). So the hyperplane is the set of all points $B$ from $\mathbb{R}^n$ such that $\overrightarrow{AB}\ $ is a linear combination of the given $n-1$ vectors. Is my understanding of a hyperplane correct?
But it seems here they use the term plane in the sense of an object of 2 dimensions determined by a point in $\mathbb{R}^n$ (the point $A = h(s)$) and by two linearly independent vectors $T(s), N(s)$. Is that the right way to understand item [iv]? I mean, is it the set of all points $B$ such that the $\overrightarrow{AB}\ $ is a linear combination of $T(s), N(s)$?
Also, is my understanding/definition of a hyperplane correct?
Sorry about the math pictures, I am not sure I can type all this in MathJax.
Note that Fig.2.17 (i) doesn't help to clear my confusion because in that case $n=3$ i.e. $n-1 = 2$ so there's no difference between a hyperplane and a plane (at least in the way which I currently understand them).