Vectors & Matrices Defined Up to Scale

[gnome]

What does it mean for a matrix or a vector to be "defined up to scale"?

I don't remember ever seeing this expression in my very limited exposure to linear algebra. To put it in context, I'm finding it in a text on computer vision in the section on geometric camera models.
They're talking about using something called homogeneous coordinates to represent points, vectors and planes, and they put the equation of a plane:
$$ax + by + cz -d = 0$$
as the dot product of two vectors:
$$\Pi \cdot P = 0$$ (Eq. 2.2)
where
$$\Pi = \left ( \begin{array}{c} a\\ b\\ c\\ -d \end{array} \right ) \; \text{and} \; P = \left ( \begin{array}{c} x\\ y\\ z\\ 1 \end{array} \right )$$

Anyway, it goes on to say

Note that $\Pi$ is only defined up to scale since multiplying this vector by any nonzero constant does not change the solutions of Eq. 2.2. We use the convention that homogeneous coordinates are only defined up to scale, whether they represent points or planes...

I don't understand what is accomplished by putting the equation in this form, nor do I understand the significance of "defined up to scale". I don't see this terminology anyplace in my (elementary) linear algebra textbook (or I don't know what to look for). Any idea where I can find a clear explanation?


(Why did I write "matrix question" as the title of this thread? I only mentioned vectors, not matrices, in my question, but this section of the text also deals with many matrices that are "only defined up to scale".)

 

[HallsofIvy]

In "homogenous coordinates", two "vectors", where one is a multiple of the other, are considered to be the same vector.

That is, <1, 2, 3, 1> , <2, 4, 6, 2>, and <3, 6, 9, 3> are all different representations of the same (3 dimensional) vector. In terms of ordinary 4 dimensional coordinates, they would, of course, represent vectors having the same direction but different lengths- hence, the three dimensional vector they all represent is "defined up to scale".

 

[gnome]

Thanks HoI. I was looking in all the wrong places. I see there's quite a bit of info on "homogenous coordinates" on the web.