10.1 Cartesian coordinates

Coordinates convert a picture or physical location into numbers. They let you describe positions, compare locations, and state which points satisfy a constraint. This is the base for distance, vectors, and geometric optimization.

This is foundational geometry. Begin with OpenStax's coordinate-plane lesson. You only need signed numbers and simple inequalities.

A basic idea: ordered positions

A point $(x,y)$ is located by moving $x$ units horizontally and $y$ units vertically from the origin $(0,0)$. Positive $x$ points right; positive $y$ points up. Order matters: $(2,3)$ and $(3,2)$ are different points.

A quick tour of properties

The axes divide the plane into four quadrants, numbered I to IV anticlockwise from the top right: I has $x>0,y>0$; II has $x<0,y>0$; III has $x<0,y<0$; IV has $x>0,y<0$. A point on an axis has at least one coordinate equal to zero (the origin has both). Between $A=(x_1,y_1)$ and $B=(x_2,y_2)$, the horizontal and vertical changes are $\Delta x=x_2-x_1$ and $\Delta y=y_2-y_1$. The midpoint is $\bigl((x_1+x_2)/2,(y_1+y_2)/2\bigr)$. An inequality such as $x+y\le4$ includes its boundary line $x+y=4$ and every point on one side of it (here, the side containing the origin).

Application 1: a robot's position

A robot starts at $(-2,1)$ and moves five units right and three units down. Where does it finish?

1. Add $5$ to the horizontal coordinate: $-2+5=3$.
2. Subtract $3$ from the vertical coordinate: $1-3=-2$.
3. Its final position is $(3,-2)$, in the fourth quadrant.

Application 2: a feasible location

A point must satisfy $x\ge0$, $y\ge0$, and $x+y\le4$. Is $(1,2)$ feasible? Is $(3,3)$?

For $(1,2)$, both coordinates are nonnegative and $1+2=3\le4$, so it is feasible. For $(3,3)$, $3+3=6>4$, so it is outside the region. The boundary $x+y=4$ is included because the sign is $\le$.