Can we have normal vectors to a plane that do not start at the origin?

Question

I am studying Siefken's Linear Algebra and came across the following example on page 126:

The underlined section doesn't quite make sense to me. I think there are many normals to $\mathcal{P}$ that do not get sent to $\vec 0$ when $T$ is applied. For example, the vector shown in the graph below, which would map to $A$.

Can someone explain what is going on here? I think there's something confusing me, but I'm not sure what it is. I'm no longer sure that the entity displayed above is a vector. Can we have normal vectors to a plane that do not start at the origin?

Answer 1

You are confusing vectors and points.

Geometrically if you project a normal vector onto the plane you get a point. You mistakenly think about it as being the vector starting into the origin and ending at this point.

If you project a vector not starting at the origin onto the plane, what you get is the vector between the projections of the start point and end point. In this case, you get a vector between a point and the same point, meaning the zero vector.

Answer 2

Projecting a vector onto a plane or another vector means taking a dot product. If the dot product is zero, you have a normal vector to the plane or vector. T is a projection transformation, so taking T on all normal vectors to the plane P will bring you the zero vector since dotting two vectors that are normal to each other, as explained before, will give you zero.

In the example that you have provided, projecting that vector onto the plane just gives you back the point that it started on. The difference between the original point where the vector stood, and the final point is zero since you ended up at the same point.

You can think of the projection as "squishing" the vector into the plane and analyzing where the resultant vector ends up. Doing it to any normal vector, you can intuitively tell that you "squish" the vector right into the point you started on.