Say I have two rectangular ratios. One rectangle's height is one and an eighth times its width, and another rectangle's height is one and three fifths times its width. I wish to find whether or not certain numbers of rectangles of the first ratio can tile a single rectangle of the second ratio from edge to edge.
It would be really neat if there were a way to do this using a linear algebra operation on four vector lengths.
If we define $r_{1}$ as the rectangle with height $h_{r_{1}}=\frac{9w_{r_{1}}}{8}$ and width $w_{r_{1}}$ and $r_{2}$ as the rectancle with height $h_{r_{2}}=\frac{8w_{r_{2}}}{5}$ and width $w_{r_{1}}$, then $A_{r_{1}} = \frac{9 w_{r_1}^2}{8}$ and $A_{r_{2}}=\frac{8 w_{r_2}^2}{5}$
The only way $r_{1}$ can tile $r_{2}$ without any remainder is if $A_{r_{2}} > A_{r_{1}} \implies A_{r_{2}} = A_{r_{1}}x$
We can solve the equation for $x$ and we get $x = \frac{64 w_{r_2}^2}{45 w_{r_1}^2}$
Furthermore, if $w_{r_{1}} > 0 \land w_{r_{2}}>0$, then $w_{r_1}>0\land w_{r_2}>\frac{3}{8} \sqrt{5} \sqrt{w_{r_1}^2}$
The answer really depends on the width of the rectangles: $r_1$ can tile $r_2$ $x$ times