Three Dimensional Coordinate Systems

Three coordinate planes

上下左右可推广出八个平面

\(\mathbb{R}\times\mathbb{R}\times\mathbb{R}=\{(x,y,z)\mid x,y,z\in\mathbb{R}\}\), \({\displaystyle \mathbb R^3}\) is the set of all ordered triples of real numbers, called three dimensional rectangular coordinate system.

Surfaces

FIGURE 9 : with no restriction on \(z\)

Distance and Spheres

Distance Formula in Three Dimensions : The distance \(\left|P_1P_2\right|\) between the points \(P_{1}(x_{1},y_{1},z_{1})\) and \(P_{2}(x_{2},y_{2},z_{2})\) is

\[ \begin{vmatrix}P_1P_2\end{vmatrix}=\sqrt{(x_2-x_1)^2+(y_2-y_1)^2+(z_2-z_1)^2} \]

Equation of a Sphere : An equation of a sphere with center \(C(h,k,l)\) and radius \(r\) is

\[ (x-h)^2+(y-k)^2+(z-l)^2=r^2 \]

In particular, if the center is the origin \(O\), then an equation of the sphere is

\[ x^2+y^2+z^2=r^2 \]