Coordinate Space, Cartesian Coordinate System
Definition
When the coordinate plane and the number line meet at the origin of the coordinate plane and are drawn orthogonally, it is called a coordinate space. The vertical line that is orthogonal to the coordinate plane is called the $z-$axis. The point determined by the three axes as shown above is called point $(a,b,c)$. Point $(0,0,0)$ is called the origin.
- The coordinate plane made by the $x$ axis and the $y$ axis is called the $xy-$ plane.
- The coordinate plane made by the $y$ axis and the $z$ axis is called the $yz-$ plane.
- The coordinate plane made by the $z$ axis and the $x$ axis is called the $zx-$ plane.
Description
Named after Descartes, who is known as the first person to conceive this, it is also called the (3-dimensional) Cartesian coordinate system.
Unit Vector
The unit vectors of the Cartesian coordinate system are as follows.
$$ \begin{align*} \hat{\mathbf{x}} &= \hat{\mathbf{x}}_{1} = \mathbf{i} = (1,0,0) \\ \hat{\mathbf{y}} &= \hat{\mathbf{x}}_{2} = \mathbf{j} = (0,1,0) \\ \hat{\mathbf{z}} &= \hat{\mathbf{x}}_{3} = \mathbf{k} = (0,0,1) \\ \end{align*} $$
Therefore, any point in the coordinate space $(x, y, z)$ is represented as follows.
$$ (x, y, z) = x\mathbf{i} + y\mathbf{j} + z\mathbf{k} $$
Relationship with Spherical Coordinate System
Coordinate Transformation
When expressing 3D Cartesian coordinates with the spherical coordinate $(r, \theta, \phi)$, from the definition of trigonometric functions, it is as follows.
$$ \begin{align*} x &= r \sin\theta \cos\phi \\ y &= r \sin\theta \sin\phi \\ z &= r \cos\theta \end{align*} $$
Conversely, expressing spherical coordinates with Cartesian coordinates is as follows.
$$ \begin{align*} r &= \sqrt{x^{2} + y^{2} + z^{2}} \\ \theta &= \cos^{-1}\textstyle\frac{z}{\sqrt{x^{2} + y^{2} + z^{2}}} \\ \phi &= \tan^{-1}\textstyle\frac{y}{x} \end{align*} $$
Derivation
$$ \dfrac{y}{x} = \dfrac{r \sin\theta \sin\phi}{r \sin\theta \cos\phi} = \tan\phi \implies \phi = \tan^{-1}\dfrac{y}{x} $$
$$ \cos\theta = \dfrac{z}{r} = \dfrac{z}{\sqrt{x^{2} + y^{2} + z^{2}}} $$