Write the equations of the edges of the tetrahedron A(0,0,2)…
Write the equations of the edges of the tetrahedron $A(0,0,2)$, $B(4,0,5)$, $C(5,3,0)$, $D(-1,4,2)$.
Explanation of Method
The equations of the six edges of a tetrahedron.
Step 1: Method
Each edge is the line through two vertices: $\frac{x-x_1}{x_2-x_1}=\frac{y-y_1}{y_2-y_1}=\frac{z-z_1}{z_2-z_1}$.
A zero denominator means that coordinate is constant.
Step 2: The six edges
| Edge | Direction | Equations |
|---|---|---|
| $AB$ | $\{4,0,3\}$ | $\tfrac x4=\tfrac{z-2}{3},\ y=0$ |
| $AC$ | $\{5,3,-2\}$ | $\tfrac x5=\tfrac y3=\tfrac{z-2}{-2}$ |
| $AD$ | $\{-1,4,0\}$ | $\tfrac x{-1}=\tfrac y4,\ z=2$ |
| $BC$ | $\{1,3,-5\}$ | $\tfrac{x-4}1=\tfrac y3=\tfrac{z-5}{-5}$ |
| $BD$ | $\{-5,4,-3\}$ | $\tfrac{x-4}{-5}=\tfrac y4=\tfrac{z-5}{-3}$ |
| $CD$ | $\{-6,1,2\}$ | $\tfrac{x-5}{-6}=\tfrac{y-3}1=\tfrac z2$ |
Check: the vertices are not coplanar, so the solid is a genuine tetrahedron: \[\big[\overrightarrow{AB},\overrightarrow{AC},\overrightarrow{AD}\big]=\begin{vmatrix}4&0&3\\5&3&-2\\-1&4&0\end{vmatrix}=101\ne0,\qquad V=\tfrac{101}{6}.\]
Result
\[\begin{aligned} AB&:\ \tfrac x4=\tfrac{z-2}{3},\ y=0\\ AC&:\ \tfrac x5=\tfrac y3=\tfrac{z-2}{-2}\\ AD&:\ \tfrac x{-1}=\tfrac y4,\ z=2\\ BC&:\ \tfrac{x-4}1=\tfrac y3=\tfrac{z-5}{-5}\\ BD&:\ \tfrac{x-4}{-5}=\tfrac y4=\tfrac{z-5}{-3}\\ CD&:\ \tfrac{x-5}{-6}=\tfrac{y-3}1=\tfrac z2 \end{aligned}\]