Help with Proposition $2.3.3$ from Elem. Differential Geometry by Pressley

Question

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Why can we have $\mathbf v \cdot \mathbf N =d$? Why is $\mathbf v \cdot \mathbf N =d$ a plane?

Where did $\gamma \cdot \mathbf N=d$ come from? Why can we do this?

Answer 1

Since I don't know how to post drawings of vectors you'll have to do the drawing for me. Draw a vector with any orientation you like and call it $\mathbf N$. Consider the point at the tail of the vector and call it $\mathbf p$. Consider all vectors of all lengths whose tails are at $p$ and are perpendicular to $\mathbf N$.

Can you see that this set of vectors forms a plane? If not, take two pencils and hold one fixed and the other at 90 degrees. Rotate the non-fixed one around, keeping it at 90 degrees. It will sweep out a disk. Now, if that rotating pencil could change lengths you'll see that this forms a plane.

Let's take this idea and derive the expression for the plane. The vectors that make up the plane are all of the form $\mathbf v - \mathbf p$. These vectors are also perpendicular to $\mathbf N$ which means that $$\mathbf N\cdot(\mathbf v - \mathbf p)=0\implies \mathbf N\cdot\mathbf v = \mathbf N\cdot\mathbf p$$.

$\mathbf N\cdot\mathbf p$ is a real number, call it $d$. Then $$\mathbf N\cdot\mathbf v=d$$ which we have already seen is a plane.