📕 The Coordinate Plane

📗 Cartesian Plane

COORDINATE PLANE is a flat surface formed by the intersection of two perpendicular lines or AXES:

• The horizontal line, known as the X-AXIS, or abscissa.
• The vertical line, known as the Y-AXIS, ordinate.
• Origin: The x- and y-axes intersect (cross) at the ORIGIN, where all lines take the $0$ value.

y
│⠀⠀⡘
│⣀⠔
┼────x
0̂
Origin

An ordered pair gives the coordinates (exact location) of a POINT. They are called an “ordered pair” because the order matters. The x-coordinate always comes first, then the y-coordinate, like so : $( x,y )$. The x- and y- coordinates are separated by a comma and surrounded by parentheses.

EXAMPLE: The x-coordinate of the origin is 0, and the y-coordinate of the origin is also 0. So, the ordered pair of the origin is $(0, 0)$.

📘 Navigating the Coordinate Plane

• If $x > 0$, POSITIVE, move RIGHT.
• If $x < 0$, NEGATIVE, move LEFT.
• If $x = 0$, STAY on the origin of x.
• If $y > 0$, POSITIVE, move UP.
• If $y < 0$, NEGATIVE, move DOWN.
• If $y = 0$, ZERO, STAY on the y-origin.

📙 Distance on the Coordinate Plane

You can use this $\text{distance formula}$ to find the line-length of the :

\color{#F00}
d²=
( x₂ - x₁ )² + ( y₂ - y₁ )²

where:

• $(x₁, y₁)$ and $(x₂, y₂)$ are two distinct points.
• The subscripts indicate the first and second points.
• The order of points does not matter, only consistency in subtraction.

From 🟣 to 🔴.

\color{#F05}
d²= ∑ᵢ ( 🔴·î - 🟣·î )² 

📗 Intervals and Variables

A variable $x_i$ can have valid values inside a given interval.

Interval:
┈─╂──╂──╂──╫──╂──╂──╂──╂──╂──╂──╂─┈

The máximum and minimum values of an interval are called the extremes.

Once a variable has a value in the interval we represent that value with a mark in the interval in the spot of that value.

x = 4
┈─╂──╂──╂──╫──╂──╂──╂──🔴─╂──╂──╂─┈

We can represent an ordered pair if we cross perpendicularly two intervals, one for each variable.

(x,y) = (4,2)
┿ y
┿
┿     2
┿╴╴╴╴╴╴╴╴╴╴╴🔴
┿           ╎4
╬──╂──╂──╂──╂──╂──╂──╂─┈ x

📘 Polar Coordinates

That same point can be expressed as:

• A point at $r$ distance to the origin
• At angle $𝜃$ to a given angle of origin, usually $\hat{x}$. According to the context, this angle is expressed in radians or degrees.

📙 System Conversion (Cartesian ⟷ Polar)

As we've seen, the same point can be determined in two different ways. We can transform from one to the other, as they are analogous. A point can be converted between Cartesian $(x, y)$ and Polar $(r, 𝜃)$ coordinates using these formulas:

\color{#F66}
x = r · cos 𝜃

\color{#F86}
y = r · sin 𝜃

\color{#68F}
r² = x²+ y²

\color{#66F}
tan 𝜃 = y : x  = \frac{y}{x}

These conversions allow movement between rectangular and polar representations, depending on the context and application.

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Understanding the coordinate plane is fundamental to mathematics and its applications in physics, engineering, and computer science. Whether using Cartesian coordinates for plotting or polar coordinates for rotational motion, mastering these concepts unlocks deeper insights into spatial relationships and problem-solving.