Last week, in Fundamental Understanding of Mathematics LVIII, we saw how multiplying all the coordinates of a triangle increased the size of the triangle. This is called "scalar" multiplication, since it changes the size, or "scale" of the triangle. It only changes the size, though, and doesn't affect the slopes of any of the sides, nor does it change the angles at the corners.
So scalar multiplication is pretty simple. It does only one thing, it does it well, and it doesn't complicate matters by having side effects.
Here is a red triangle, with corners A at (7,6), B at (10,3) and C at (4,2). You'll notice I've added another Quadrant onto the graph.
Last week the graph showed positive x numbers and positive y numbers. It looked like this
Most of the time, when someone draws a graph, this is what it looks like. So when graphs expanded to include numbers on the other side of zero, each section was called a Quadrant, and this section was called the First Quadrant. Traditionally, it is written in Roman numberals, or Quadrant I.
Quadrant I has positive x and positive y numbers, and it's boundaries are the X axis on the bottom and the Y axis on the left side. There is no top boundary, nor right side boundary. The Quadrant goes forever in those directions, although we rarely draw all of it. The "goes-in-that-direction-forever" idea is included in the graph by putting that little arrowhead on the X axis pointing off to the right, and on the Y axis by that little arrowhead pointing up.
In today's graph, we have included another Quadrant, so we will call it Quadrant II, it's traditional name.
Today, let's take a look at multiplying by a negative number, namely, negative 1, just to keep things simple. Since our graph has positive and negative x numbers on it, but only positive y numbers, we will only multiply the x coordinates of the triangle's corners, and leave the y coordinates alone. So, we are no longer doing scalar multiplication, so we cannot expect scalar multiplication results.
A will be at (7 x - 1, 6) = ( - 7, 6)
B will be at (10 x - 1, 3) = ( - 10, 3)
and C will be at (4 x - 1, 2) = ( - 4, 2)
and looks like the blue triangle
It seems like the red triangle, but it's been flipped over. It's a mirror image of the red triangle, if the Y axis was the mirror.
We could do something similar by only multiplying the Y coordinates by -1, and as you'll imagine, the new triangle would also be a mirror image, but this time, with the X axis acting as the mirror.
Have fun in the comments.