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Section 11.2 Extending to higher dimensions
Objectives
-
Extend familiar geometric ideas from two dimensions to three dimensions.
-
Describe points, distances, surfaces, and regions using three coordinates.
-
Interpret equations in two variables as three-dimensional cylinders with one free variable.
-
Taking familiar concepts from 2D and extending them to 3D
-
3D coordinates:
\((x,y,z)\)
-
Midpoint:
\begin{equation*}
M=\qty(\frac{x_{1}+x_{2}}{2},\frac{y_{1}+y_{2}}{2},\frac{z_{1}+z_{2}}{2})
\end{equation*}
-
Distance:
\begin{equation*}
d=\sqrt{\Delta x^{2}+\Delta y^{2}+\Delta z^{2}}
\end{equation*}
-
Equation of a plane:
\begin{equation*}
ax+by+cz=d
\end{equation*}
-
Sphere:
\begin{equation*}
(x-x_{0})^{2}+(y-y_{0})^{2}+(z-z_{0})^{2}=r^{2}
\end{equation*}
-
Interior, exterior, boundary (intuitively)
-
Open ball:
\begin{equation*}
(x-x_{0})^{2}+(y-y_{0})^{2}+(z-z_{0})^{2}\lt r^{2}
\end{equation*}
-
Closed ball:
\begin{equation*}
(x-x_{0})^{2}+(y-y_{0})^{2}+(z-z_{0})^{2}\le r^{2}
\end{equation*}
-
Cylinders: Take an equation in two variables and think of the third variable as ‘free’
