Surfaces described using norms and dot products

Question

Okay this seems like a simple question, I think I'm just missing something obvious...

The question asks to identify surfaces from the following formulae:

1. $\vert{\bf r}\vert = a$,

2. ${\bf r}\cdot{\bf n} = b$,

3. ${\bf r}\cdot{\bf n} = c\,\vert{\bf r}\vert$,

4. $\vert{\bf r} -({\bf r}\cdot{\bf n})\,{\bf n}\vert = d$.

So the first one seems like it should obviously be a sphere, but the others I'm not sure, and I'm not sure how to approach the problem, could someone please point me in the right direction?

Answer 1

To identify the second one, let $\pmb r_0 \in \def\R{\mathbf R}\R^d$ be arbitrary with $\pmb r_0 \cdot \pmb n = b$, then (2) can be rewritten as $$ \def\r{\pmb r}\def\n{\pmb n}(\r - \r_0) \cdot \n = 0 $$ that is, (2) contains all points $\r\in \R^d$ for which $\r - \r_0$ is perpendicular to $\n$, that is, (2) describes a hyperplane (if $\n \ne 0$).

For (3), lets write $\r \cdot \n$ as $\def\abs#1{\left|#1\right|}\r \cdot \n = \abs \r \abs \n \cos\angle(\r, \n)$, that is, (3), can be written as $$ \abs \n \cdot \cos\angle(\r, \n) = c\tag{3'} $$ that is (3) consists of all points for which the angle to some given point $\n$ is fixed, that is (3) describes a cone.

For (4), note that $\r \mapsto (\r\cdot \n) \n$ is some projection onto $\R \n$ (the orthogonal one iff $\abs\n = 1$). That is (4) describes the set of points for which some (the orthogonal) distance to $\R\n$ is constant, that is a cylinder (here, the product of a $(d-1)$-sphere and a line).

Answer 2

Here's an approach: for problem 2, pick specific values for $\mathbf n$ and $b$, like $$ \mathbf n = [0,0,1]^t \\ b = 2 $$ and see what the result looks like. Then see whether you can generalize. The same approach should work for all the others.