very basic question about coordinate geometry

Question

I am taking a beginner course in machine learning and have confused myself horribly about something. The idea is that there is a line which splits a 2D plane into two distinct regions as shown in the figure:

Now the instructor says that anything below the line is greater than $0$ and anything above is less than $0$. Now, it is not clear to me why that should be? Is there an intuition behind why anything below this line should be greater than zero (according to the line equation) and anything above should be less than 0.

Answer 1

The equation $$z=1\cdot x-1.5\cdot y$$ is that of a plane. It intersects the horizontal plane $z=0$ along a line. Obviously, the plane is split in two halves, one above and one below the horizontal plane.

Answer 2

The equation of the line is $2x-3y=0$. Therefore it divides the plane into two parts, the part $2x-3y> 0$ (below the line) and the part $2x-3y< 0$ (above the line). Here $x$ is your "awesome" and $y$ is your "awful".

This is true for any line: a line $ax+by+c=0$ divides the plane into the part $ax+by+c> 0$ and the part $ax+by+c< 0$.

Answer 3

In the picture, the grey line divides the whole plane into 2 parts. The red dot A [with co-ordinates = (4, 1)] lies in the lower portion of the plane. That is why we say it lies below the line.

When we substitute the co-ordinates of A in, we have $1.0(4) – 1.5(1) = 2.5 \gt 0$.

A(4, 1) can be considered as a representative because all points that lie below the line have the same characteristics of "1.0 (awesome) – 1.5 (awful) $\gt 0$". We can therefore call the whole lower portion (region) of the plane as $$1.0 (awesome) – 1.5 (awful) \gt 0$$

Try the yellow dot and see the outcome.

Answer 4

By the very definition of the equation of a straight line, the points on the line are exactly those for which the expression is zero. In other terms, the points outside the line make the expression either positive or negative.

As the expression is continuous (it varies without jumps), if you move in the plane the value of the expression will not change sign without passing through zero. Hence the line splits the plane in two regions of opposite signs.

Answer 5

Instead of $1.0\#awesome-1.5\#awful=0$ I prefer $1.0x-1.5y=0$, which multiplying by two gives $2x-3y=0$.

Because you'd like to see which is above the line look at the term with the $y$, namely $-3y$, and see the sign. As the sign is negative then which stands above the line are the points $(x,y)$ such that $2x-3y<0$. This is because the bigger positive values of $y$'s you have the smaller negative values of $-3y$ you get. In this case your instructor says that anything above the line is negative.

If instead you use the equation $-2x+3y=0$ which produces the same line then the sign of the term with the $y$ is positive, which means that the points that are above the line are those $(x,y)$ such that $-2x+3y>0$. Again, this is because the bigger positive values of $y$ you have the bigger positive values you get. In this case your instructor would say that anything above the line is positive.