This is in continuation of my earlier post here.
On pg.#7-8, question #7 is about an alternate model of growth (i.e., proportional growth, herewith referred as P) and its comparison with the uniform growth model (herewith referred as U) used to form one rectangle from another by shrinking or expanding $x,y$ coordinates. The different parts of the question are attempted below, but am clueless about the (c), (d) parts:
(a) Asks to compare P with U in question #5(a) -(d). (There is an erratum there, refers to question #7 rather than question #5). The U model is covered in a previous post here. Below will attempt using the P model:
a) Describe all the points in the lh-plane that represent the
rectangles proportionally grown from the rectangle $(3,2)$.
-> $(3*i, 2*i), i\in \mathbb{R}, i\ne 0$
b) Given the rectangle $(3,2)$ and the rectangle $(5,3)$, can you find a rectangle that can be proportionally grown from each of them? Explain.
-> Need find if possible for real $s,t$ s.t. $(i) 3s = 5t\implies \frac{s}{t} = \frac53; 2s = 3t\implies \frac{s}{t} = \frac32$. No, it is not possible.
c) If one rectangle is proportionally grown from another, can
they be similar? Describe all such pairs of rectangles.
-> Need show that some real $s$ and given point $(a,b), \frac{a}{b} = \frac{as}{bs}$. It can be seen that it is an axiom for any value of $s$.
d) Is it possible to grow (or shrink) uniformly a square from
any rectangle? Explain.
-> Need show that for a point $(a,b)$ with $a\ne b$, there exists a real $s$ s.t. $as= bs$. No, it is always false.
(b) What is the relation between proportional growth and
aspect ratio?
No change in aspect ratio.
(c) Consider two rectangles (two points in the lh-plane, for example, A & B).
What if you are allowed to use two growth processes—one proportional and one
uniform—in sequence? Can you always grow one rectangle from another when you are allowed to use at most two steps?
(d) Under which conditions on the dimensions of two rectangles, you can grow one rectangle from another by using both the uniform and proportional growth? What can be said about the number of times you must switch from one type of growth to another one? Does the order in which you apply them matter?
For part $(a)$, you might like to impose the condition that $i > 0$.
Guide for part $(c)$,
A $P_p$ operation maps a point from $(x,y)$ to $(px,py)$ where $p>0$.
A $U_u$ operations maps a point from $(x,y)$ to $(x+u, y+u)$.
$U_u(P_p(x,y))=U_u(px,py)=(px+u, py+u)$
Given positive numbers, $x,y,r$ and $s$, can we solve for $px+u=r$, $py+u=s$.
$$\begin{bmatrix} x & 1 \\ y & 1\end{bmatrix}\begin{bmatrix} p \\ u\end{bmatrix} = \begin{bmatrix} r \\ s\end{bmatrix} $$
Suppose $x \ne y$, then we have $$\begin{bmatrix} p \\ u\end{bmatrix} = \frac{1}{x-y}\begin{bmatrix} 1 & -1 \\ -y & x\end{bmatrix}\begin{bmatrix} r \\ s\end{bmatrix} $$
Try to pick positive $x,y,r,s$ such that $p$ is negative and hence it can't be done in the order of $U$ and then $P$.
For the other order, note that we have $$P_q(U_v(x,y))=P_q(x+v,y+v)=(qx+vq, qy+vq)=U_{qv}(P_q(x,y))$$