Cartesian coordinates

Basics

Cartesian plane denoted ${\text{R}}^2$

Axes and Quadrants

$$\begin{array}{rl}x-axis& =\{(x,y)\in {\text{R}}^2:y=0\}\\ y-axis& =\{(x,y)\in {\text{R}}^2:x=0\}\\ firstquadrant& =\{(x,y)\in {\text{R}}^2:x>0,y>0\}\\ secondquadrant& =\{(x,y)\in {\text{R}}^2:x<0,y>0\}\\ thirdquadrant& =\{(x,y)\in {\text{R}}^2:x<0,y<0\}\\ fourthquadrant& =\{(x,y)\in {\text{R}}^2:x>0,y<0\}\end{array}$$

Calculating distance

Pythagoras

$$z=\sqrt{x^2+y^2}$$

Distance formula

Given three points:

$$\begin{array}{rl}A& =(x_a,y_a)\\ B& =(x_b,y_b)\\ C& =(x_c,y_c)\\ \\ dist(A,B)& =\sqrt{(x_a-x_b{)}^2+(y_a-y_b{)}^2}\\ \\ dist(A,C)& =\sqrt{(x_a-x_c{)}^2+(y_a-y_c{)}^2}\end{array}$$

Clustering

If $A$ and $B$ are in Cluster $I$
and $C$ is in Cluster $II$

Then

$dist(A,B)<<dist(A,C)$

Lines

Slope of line segment

Given
$A=(a,b)$
$B=(c,d)$

$$\begin{array}{rl}Slopeof\overrightarrow{AB}& =\\ m& =\frac{d-b}{c-a}\end{array}$$

Basic line formula

$y-y_0=m(x-x_0)$ "Point-slope" form
$y=mx+b$ "slope-intercept" form

Given a line that passes through $(2,1)$ and $(3,2)$, i.e. a slope of 1,
for any point $(x,y)$ to be on the line, the slope from $(2,1)$ to $(x,y)$ must be $1$

$$\begin{array}{rl}1& =\frac{y-1}{x-2}\end{array}$$

rearranging...

$$\begin{array}{rl}y-1& =1(x-2)\end{array}$$

so a definition of the line is:

$$\begin{array}{rl}\ell =\{(x,y)\in {\text{R}}^2:y-1& =1(x-2)\}\end{array}$$

Point-slope formula for line

Above generalises to

If a line $\ell$ has slope $m$, and if $(x_0,y_0)$ is any point on $\ell$, then $\ell$ has the equation:

$$\begin{array}{r}y-y_0=m(x-x_0)\end{array}$$

Slope-intercept formula for line

Considering same line defined as

$$\begin{array}{rl}\ell =\{(x,y)\in {\text{R}}^2:y-1& =1(x-2)\}\end{array}$$

This line intercepts the $y$-axis at a point $(0,b)$ i.e.

$$\begin{array}{rl}b-1& =1(0-2)\\ b& =-1\end{array}$$

Substituting this intercept point back into the Point-Intercept formula:

$$\begin{array}{rl}y-(-1)& =1(x-0)\\ y+1& =x\\ y& =1x-1\end{array}$$

This generalises into the Slope-Intercept formula for the line:

If $\ell$ has slope $m$, and $\ell$ hits the y-axis at $(0,b)$, then

$$\begin{array}{rl}y& =mx+b\end{array}$$

is an equation for $\ell$, where $m$ is the slope and $b$ is the y-intercept.

Based on course material by Paul Bendich and Daniel Egger from Data Science Maths Skills