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Section 3 Key Concepts

Subsection 3.1 The Standard Three-Dimensional Coordinate System—Rectangular Coordinates

With your right hand, point your thumb up, and your index and middle fingers outward perpendicular to each other. Next draw three lines, on in each direction your thumb and fingers are pointing and label them as follows:

  • the line along your thumb is the \(\displaystyle z \)-axis
  • the line along your middle finger is the \(\displaystyle y \)-axis
  • the line along your index finger is the \(\displaystyle x \)-axis

Distances with the standard three dimensional coordinate system:

  • The distance between \(P_{1}=(x_1,y_1,z_1)\) and \(P_{2}=(x_{2},y_{2},z_{2})\text{,}\) is \(|P_{1}P_{2}|=\sqrt{ (x_{1}-x_{2})^{2} +(y_{1}-y_{2})^{2} +(z_{1}-z_{2})^{2}}.\)

Because of this, an equation for the sphere with center \((a,b,c)\) and radius \(R\) is \((x-a)^2+(y-b)^2+(z-c)^2=r^2.\)

Subsection 3.2 Other coordinate systems

Polar Coordinates.

The polar coordinates of a point \(P\) are \((r,\theta)\) where \(r\) is the distance of \(P\) from the origin (\(r\geq 0\)) and \(\theta\) is the angle made with the line connecting \(P\) to the origin and the positive \(x\)-axis (\(0\leq \theta \leq 2\pi\)).

  • The rectangular coordinates of \((r,\theta)\) are \(x=r\cos (\theta )\) and \(y=r\sin (\theta )\)
  • The polar coordinates of \((x,y)\) are \((r, \theta)\) where \(r=\sqrt{x^2+y^2}\) and \(\theta\) is the solution of \(\tan (\theta )=y/x\) that is in the correct quadrant.
Cylindrical Coordinates.

The cylindrical coordinates of a point \(P\) in \({\mathbb R}^3\) are \((r,\theta ,z)\) where \((r,\theta)\) are the polar coordinates of the projection of \(P\) in the \(xy\)-plane and \(z\) is the \(z\)-coordinate.

  • The rectangular coordinates of \((r,\theta ,z)\) are

    \begin{align*} x\amp =r\cos{(\theta )},\\ y\amp =r\sin{(\theta )},\text{ and }\\ z\amp =z. \end{align*}

  • The cylindrical coordinates of \((x,y,z)\) are \((r,\theta,z)\) where

    \begin{align*} r\amp=\sqrt{x^2+y^2},\\ \theta \amp = \text{ the solution of } \tan (\theta )=y/x \text{ that is in the correct quadrant,}\\ z\amp =z. \end{align*}

Spherical Coordinates.

The spherical coordinates of a point \(P\) in \({\mathbb R}^3\) are \((\rho , \theta ,\phi )\) where \(\rho\geq 0\) is the distance of \(P\) from the origin, \(\theta\) is the same as in cylindrical coordinates, and \(\phi\) is the angle between the positive \(z\)-axis and the line segment connecting \(P\) to the origin (\(0\leq \phi \leq \pi\)).

  • The rectangular coordinates of \((\rho,\theta ,\phi)\) are

    \begin{align*} x\amp =\rho \sin{(\phi )} \cos{(\theta )},\\ y\amp =\rho \sin{(\phi )} \sin{(\theta )},\text{ and }\\ z\amp =\rho \cos{(\phi )}. \end{align*}

  • The spherical coordinates of the point \((x,y,z)\) can be found by solving the equation \(\rho^2=x^2+y^2+z^2\text{,}\) and then using the equations given above to first solve for \(\phi\) and then \(\theta\text{.}\)

Distances in other coordinate systems.

It is often easier to convert to rectangular coordinates to find distances in \(3\)-space.

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