Answer to Question #248901 in Analytic Geometry for Markzonny

Question

Use the tetrahedron that has corner coordinates A=(1,-1,0), B=(6,1,3), C= (–3,2,0) and D=(0,6,5) and  check that its centroid will be at (1,2,2).

a)A tetrahedron has 4 corners, 4 faces, 6 edges, and 12 corner angles. How many of our tetrahedron’s 12 corner angles are right angles, and which one(s)?

(b) Determine the projection of the tetrahedron onto the viewing screen when the eye is at (–6,2,2). Colour the side or sides that the viewer can see from this position.

Answer 1

The centroid is the average of the vertices:

$\overline{x}=\frac{x_1+x_2+x_3+x_4}{4}=\frac{1+6-3}{4}=1$

$\overline{y}=\frac{y_1+y_2+y_3+y_4}{4}=\frac{-1+1+2+6}{4}=2$

$\overline{z}=\frac{z_1+z_2+z_3+z_4}{4}=\frac{3+5}{4}=2$

a)

for perpendicular vectors dot product = 0

we have:

$AB=(5,2,3),BC=(-9,1,-3),CD=(3,4,5)$

$AC=(-4,3,0),AD=(-1,7,5),BD=(-6,5,2)$

$AB\cdot BC=-45+2-9\neq 0$

$AB\cdot AC=-20+6\neq 0$

$AB\cdot AD=-5+14+15\neq 0$

$AB\cdot BD=-30+10+6\neq 0$

$BC\cdot CD=-27+4-15\neq 0$

$BC\cdot AC=36+3\neq 0$

$BC\cdot BD=54+5-6\neq 0$

$CD\cdot AC=-12+12=0$

$CD\cdot AD=-3+28+25\neq 0$

$CD\cdot BD=-18+20+10\neq 0$

$AC\cdot AD=4+21\neq 0$

$AD\cdot BD=6+35+10\neq 0$

So, there is one right angle $\angle ACD$

b)

the eye is at point O

the viewer can see faces ABC and BCD