logo

Coordinate Space, Cartesian Coordinate System 📂Mathematical Physics

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−z-axis. The point determined by the three axes as shown above is called point (a,b,c)(a,b,c). Point (0,0,0)(0,0,0) is called the origin.

  • The coordinate plane made by the xx axis and the yy axis is called the xy−xy- plane.
  • The coordinate plane made by the yy axis and the zz axis is called the yz−yz- plane.
  • The coordinate plane made by the zz axis and the xx axis is called the zx−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.

x^=x^1=i=(1,0,0)y^=x^2=j=(0,1,0)z^=x^3=k=(0,0,1) \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)(x, y, z) is represented as follows.

(x,y,z)=xi+yj+zk (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,θ,ϕ)(r, \theta, \phi), from the definition of trigonometric functions, it is as follows.

x=rsin⁡θcos⁡ϕy=rsin⁡θsin⁡ϕz=rcos⁡θ \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.

r=x2+y2+z2θ=cos⁡−1zx2+y2+z2ϕ=tan⁡−1yx \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

yx=rsin⁡θsin⁡ϕrsin⁡θcos⁡ϕ=tan⁡ϕ  ⟹  ϕ=tan⁡−1yx \dfrac{y}{x} = \dfrac{r \sin\theta \sin\phi}{r \sin\theta \cos\phi} = \tan\phi \implies \phi = \tan^{-1}\dfrac{y}{x}

cos⁡θ=zr=zx2+y2+z2 \cos\theta = \dfrac{z}{r} = \dfrac{z}{\sqrt{x^{2} + y^{2} + z^{2}}}