What is Cartesian Coordinates & Formulas?

[Greg Bernhardt]

Definition/Summary

Cartesian coordinates are ordinary rectangular coordinates in a flat Euclidean space.

Cartesian form of a complex number is the form $x + iy$, where $x$ and $y$ are real.

Cartesian product of two or more sets is the most general product set, the direct product, with the symbol $\times$.

Equations

CARTESIAN COORDINATES:

2 dimensions:

$(x,y)$ or $(x_1,x_2)$

3 dimensions:

$(x,y,z)$ or $(x_1,x_2,x_3)$

n dimensions:

$$(x_1,x_2,\cdots x_n)$$

CARTESIAN FORM:

$$x + iy$$

CARTESIAN PRODUCT (DIRECT PRODUCT):

$$X\,\times\,Y\ =\ \{(x,y)\,:\,x\in X, y\in Y\}$$

$$\prod_{i = 1}^n X_i\ =\ X_1\times X_2\times\cdots X_n\ =\ \{(x_1,x_2,\cdots x_n)\,:\,x_1\in X_1, x_2\in X_2,\cdots x_n\in X_n\}$$

$$\prod_{i\in A}^n X_i\ =\ \{f:A\to\bigcup_{i\in A}X_i\,:\,f(i)\in X_i\,,\,\forall i\}$$

Extended explanation

Inner product (dot-product):

The inner product (dot-product) of two vectors in Cartesian coordinates $(x_1,x_2,\cdots x_n)$ and $(y_1,y_2,\cdots y_n)$ is the sum:
$(x_1,x_2,\cdots x_n)\cdot(y_1,y_2,\cdots y_n)\ =\ x_1y_1\,+\,x_2y_2\,+\,\cdots x_ny_n$

Cross-product:

The cross-product of two vectors exists only in 2 and 3 dimensions:

$(x_1,x_2)\times(y_1,y_2)\ =\ x_1y_2\,-\,x_2y_1$

$(x_1,x_2,x_3)\times(y_1,y_2,y_3)\ =\ (x_2y_3\,-\,x_3y_2,x_3y_1\,-\,x_1y_3,x_1y_2\,-\,x_2y_1)$

Minkowski coordinates:

By comparison, Minkowski coordinates are also rectangular, but they are not in a Euclidean space, and so are not Cartesian.

4-dimensional space-time (Minkowski coordinates):

$$(t,x,y,z)$$ or $$(x_0,x_1,x_2,x_3)$$

$(x_0,x_1,x_2,x_3)\cdot(y_0,y_1,y_2,y_3)\ =\ x_0y_0\,-\,x_1y_1\,-\,x_2y_2\,-\,x_3y_3$

Polar coordinates:

The most common alternative to Cartesian coordinates in 2 dimensions are the polar coordinates, $(r,\theta)$:

$$x\ =\ r\cos\theta,\ y\ =\ r\sin\theta$$
$$r\ =\ \sqrt{x^2+y^2},\ \theta\ =\ tan^{-1}(y/x)$$

The most common alternatives to Cartesian coordinates in 3 dimensions are the spherical coordinates, $(r,\theta,\phi)$:

$$x\ =\ r\cos\theta\cos\phi,\ y\ =\ r\sin\theta\cos\phi,\ z\ =\ r\sin\phi$$

(some authors interchange $\theta$ and $\phi$)

and the cylindrical (or polar) coordinates $(r,\theta,z)$:

$$x\ =\ r\cos\theta,\ y\ =\ r\sin\theta,\ z\ =\ z$$

Cartesian form of a complex number:

There are two standard representations of a complex number: Cartesian and polar.

The Cartesian form is $x\ +\ iy$, where $x$ and $y$ are real.

The polar form is $re^{i\,\theta}$, where $r\ =\ \sqrt{x^2+y^2}$ is the modulus and $\theta\ =\ tan^{-1}(y/x)$ is the argument.

They are so called by analogy with 2-dimensional Cartesian and polar coordinates: complex numbers in Cartesian form may be mapped directly onto the Argand plane in Cartesian coordinates, and complex numbers in polar form may be mapped directly onto the Argand plane in polar coordinates:

$x\ +\ iy\ \to\ (x,y)$ and $re^{i\,\theta}\ \to\ (r,\theta)$

which is why the equations (above) which convert Cartesian to polar coordinates also convert Cartesian to polar form.

Polar form is useful for multiplying (or dividing) complex numbers: $re^{i\,\theta}se^{i\,\phi}\ =\ rse^{i\,(\theta +\phi)}$, but not for adding them.

Cartesian product:

The Cartesian product $X\times Y$ of two sets $X$ and $Y$ is the set of ordered pairs $(x,y)$ of which the first is in $X$ and the second is in $Y$

The Cartesian product $X_1\times X_2\times\cdots X_n$ of n sets is the set of ordered n-tuples $(x_1,x_2,\cdots x_n)$ such that $x_1\in X_1, x_2\in X_2,\cdots x_n\in X_n$

An n-tuple can also be regarded as a function from the set $\{1,2,\cdots n\}$ into the set $(X_1\bigcup X_2\bigcup \cdots X_n)$ such that $f(1)\in X_1,f(2)\in X_2,\cdots f(n)\in X_n,$

Similary, we can define an infinite Cartesian product:
$$\prod_{i\in A}^n X_i\ =\ \{f:A\to\bigcup_{i\in A}X_i\,:\,f(i)\in X_i\,,\,\forall i\}$$

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[fresh_42]

Cartesian coordinates require that a space has a orthogonal basis, i.e. at least a measure for angels, pre Hilbert spaces. For a list of categories of vector spaces in general see the graphic in https://www.physicsforums.com/insights/hilbert-spaces-relatives/

Cartesian coordinates are a special form of a Cartesian product, where the factors are arranged in an orthogonal way. So in this sense, a Cartesian product is a set theoretical construct like if we had various baskets and an element is a tuple where we have chosen one element from each basket.