回顾：什么是笛卡尔坐标？

One way is to use rectangular coordinates, which first defines a pair of perpendicular lines called the $x$ and $y$ axes:

The intersection of these lines is called the origin, commonly labeled $O$.

Now, we can define points based on their distances to the axes. For example, the point $(2,3)$ is the point we get when we go 2 units along the $x$ axis, and 3 units along the $y$ axis. Note that these distances are signed, which means each line has a positive and negative side. To see the effect of the signs, look at the positions of $(2,3)$ (blue) and $(-2,3)$ (red) on the plot below:

Do you now see how every point on the plane can be described as a pair of two numbers, $(a,b)$, where $a$ is the distance you have to travel along the $x$ axis and $b$ is the distance you have to travel along the $y$ axis?

Being able to describe points with numbers is very useful. With coordinates, we can turn geometric objects, such as lines, into equations of numbers! In the graph above, the blue line is the set of all points $(x,y)$ such that $y=x+1$. The green line is $y=\frac{x}{2}-3$, and the red line is $y=-2x-12$.

You'll see the power of this relationship between geometry and algebra a lot as you move forward!

Analytic geometry is the analysis of points and lines on a coordinate plane.

Here's a point in the plane:

Here's a line:

The great thing about analytic geometry is that it combines geometric figures and algebra together. The understanding of basics of geometry and algebra are both important as algebra is needed to analyze the movement and interaction of geometric figures in space.

It's also the base-stone of calculus, so have fun, and good luck!

The Cartesian coordinate system is a coordinate system that specifies each point uniquely in a plane by a pair of numerical coordinates, which are the signed distances from the point to fixed perpendicular directed lines, measured in the same unit of length.

In the two-dimensional case, it has two axes, the $x$- and $y$-axes and looks like this:

A point is given in the form $(x,y)$ where $x$ is the horizontal value and $y$ is the vertical value. The cartesian coordinate plane is divided into four quadrants. The top-right quadrant ($x>0$, $y>0$) is Quadrant I, and then the numbers move up as we go counter-clockwise, until the bottom-right quadrant ($x>0$, $y<0$) is Quadrant IV.

Welcome to the land of coordinate geometry, where we turn geometry problems into algebra problems (more specifically, algebraic identities like $a^2-b^2 = (a-b)(a+b)$)!

Of course, it's helpful if you know some "actual" geometry basics before learning coordinates---otherwise the algebra is meaningless. I recommend learning about the Pythagorean theorem and distance formula, for instance.

If you've already done so, let's proceed with the art of coordinate bashing!

In $2$ dimensions: The notion of Cartesian coordinates requires $2$ lines in a plane that intersect and are perpendicular to each other. Each line is called an axis. Then the location of any point in the plane is described by a pair of numbers, where each number is the perpendicular distance to one of the axes.

Typically, the axes are called the x and y axes. The point at $(3,4)$ for example, is $3$ units away from the $x$-axis and $4$ units away from the $y$-axis.