Last week, in Fundamental Understanding of Mathematics LVII, we moved a triangle drawing around on a two dimensional Cartesian coordinate system (aka, a graph) by adding an ordered pair to the coordinates of the triangle's corners. This week, we are going to see what multiplication does to our triangle
Here is our graph with a triangle drawn on it. The triangle's corners are (2,3),(5,7),(8,4).
Let's multiply everything by, say, 1.2. The new corners would be at (2x1.2, 3x1.2) or (2.4, 3.6);
(5x1.2, 7x1.2) or (6, 8.4) and (8x1.2, 4x1.2) or (9.6, 4.8). I'll draw the new triangle in blue.
The triangle seems to have gotten larger, and moved a bit. We can remove the "moved a bit" by getting the lower left corner to match up with the original lower left corner, using last week's technique. (We can do this for any corner, I just happened to pick the lower left), To move the triangle so the lower left corner matches, we find how much the lower left corner moved, and subtract that amount from all corners of the new triangle.
That is to say: the original lower left corner was at (2, 3). The new triangle's lower left corner is at (2.4, 3.6). The difference is (0.4, 0.6). (Notice how I'm using the ordered pair notation – I'm finding the difference in the first number in the pair, the x coordinate; then I'm finding the difference in the second number in the pair. I'm writing the differences in the first and second positions, since one of the differences is the x difference, and the other is the y difference.)
Now, if I subtract the difference from all the corners, I'll have slid the new triangle so the lower left corners match.
The new corners are at (2, 3); (5.6, 7.8) and (9.2, 4.2)
Seems odd that two sides of the red triangle are covered up by the slightly longer sides of the blue triangle. It's as if the angle at that lower left corner is exactly the same. It also appears that the angles at the other two corners are the same, too. If we had chosen to move the highest corner of the blue triangle to match the highest corner of the red triangle, we would see that the angle is the same, as well.
Multiplying all the coordinates by the same positive number simply makes the shape larger, without changing any of the angles.
Have fun in the comments.