Question

Alright this is a continuation of my regular tetrahedron inside a cube problem, after finding the measure of the angle between the carbon molecules and the hydrogen molecules in methane.

Use dot product to find the angle between two adjacent edges (sharing a common vertex) of the tetrahedron; and the angle between two opposite edges. Explain your answer using symmetry.

Answer 1

here is our tetrahedron \[(1, 1, 1), (-1, 1, -1), (-1, -1, 1), (1, -1, -1)\]

Answer 2

actually, those vectors represent the vertices of that tetrahedron. What do I do if I want to find vectors that represent the edges of my tetrahedron
?

Answer 3

vector addition. right?

Answer 4

what is the vector from (1,1,1) to (-1,1,-1)?

Answer 5

their difference, right?

Answer 6

yes

Answer 7

\[A := (1,1,1)\]
\[B:=(-1,1,-1)\]
\[C:=(-1,-1,1)\]
\[D:=(1,-1,-1)\]

Answer 8

\[AB = A - B = (1,1,1)-(-1,1,-1)=(2,0,2)\]

Answer 9

umm or is it the other way around? B-A?

Answer 10

(-1,1,-1)-(1,1,1)=(-2,0,-2)=AB

Answer 11

yeah it's the other way around \[AB = B-A = (-2,0,2)\]

Answer 12

\[AC = C-A=(-1,-1,1)-(1,1,1)=(-2,-2,0)\]
\[AD = D-A=(1,-1,-1)-(1,1,1)=(0,-2,-2)\]

Answer 13

those are like 3 adjacent edges and I only need 2 to solve the problem.

Answer 14

What about if I want to find the angle of 2 *opposite* edges of a tetrahedron? what are opposite edges?

Answer 15

btw AB should be \[(-2,0,-2)\]

Answer 16

hmm when you visualize it.... the angle between 2 opposite edges of a tetrahedron is pi/2 radians.

Answer 17

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