The Geometry of a Tetrahedron

1
The Geometry of a Tetrahedron

• Footnote 18Section 10.4
• Mark Jeng
• Professor Brewer

2
What is a tetrahedron?

• A tetrahedron is a solid with 4 vertices P, Q,
  R, and S.
• There are also 4 triangular faces opposite the
  vertices as shown in the figure.

3
Problem 1

• 1. Let v1, v2, v3, and v4 be vectors with lengths
  equal to the areas of the face opposite the
  vertices P, Q, R, and S, respectively, and
  direction perpendicular to the respective faces
  and pointing outward. Show that
• v1 v2 v3 v4 0

4
Setting up the vectors

• The area of a triangle is ½a x b
• v1 ½ QS x SR
• v2 ½ PS x PR
• v3 ½ PS x PQ
• v4 ½ PQ x PR

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Vector 1

• ½ i(0) j(0) k(-qr)
• v1 ½ ½ lt0,0,-qrgt
• ½ qr
• The direction of the vector is pointing outward
  from the xy-axis, so therefore the final vector
  is
• v1 lt0,0,-½ qrgt

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Vector 2

• ½ i(pr) j(0) k(0)
• v2 ½ ½ ltpr,0,0gt
• ½ pr
• The direction of the vector is pointing outward
  from the yz-axis, so therefore the final vector
  is
• v2 lt-½ pr ,0,0gt

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Vector 3

• ½ i(0) j(pq) k(0)
• v3 ½ ½ lt0,-pq,0gt
• ½ pq
• The direction of the vector is pointing outward
  from the xz-axis, so therefore the final vector
  is
• v3 lt0,-½ pq,0gt

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Vector 4

• ½ i(-pr) j(-pq) k(qr)
• v4 ½ ½ lt-pr,pq,qrgt
• ½ v(-pr)2 (pq)2 (qr)2
• The vector is pointing outward in the x, y, and
  z directions perpendicular to the plane PQR
  therefore the final vector is
• v4 lt½ pr,½ pq,½ qrgt

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Question 1 Results

• Show that v1 v2 v3 v4 0
• lt0,0,-½ qrgt lt-½ pr ,0,0gt lt0,-½ pq,0gt lt½
  pr,½ pq,½ qrgt 0
• The x, y, and z components of the vectors all
  cancel out

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Problem 2

• 2. The volume V of a tetrahedron is 1/3 the
  distance from a vertex to the opposite face,
  times the area of that face.
• (a) Find a formula for the volume of a
  tetrahedron in terms of the coordinates of its
  vertices P, Q, R, and S.
• (b) Find the volume of the tetrahedron whose
  vertices are P(1,1,1), Q(1,2,3), R(1,1,2), and
  S(3,-1,2).

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Formula for Tetrahedron Volume

• (a) The volume of a tetrahedron is
• V 1/3 d(s1, s2, s3), PlanePQR ½ PQ x PR
• where (s1, s2, s3) are the x, y, and z
• components of point S, the vertex
• opposite PlanePQR

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Finding the Volume

• (b) Find the volume using the formula
• V 1/3 d(s1, s2, s3), PlanePQR ½ PQ x PR
• with points P(1,1,1), Q(1,2,3), R(1,1,2), and
  S(3,-1,2).
• The first step is to derive the vectors PQ and
  PR
• vPQ lt1-1, 2-1, 3-1gt lt0,1,2gt
• vPR lt1-1, 1-1, 2-1gt lt0,0,1gt

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Finding the Volume (cont.)

• Using those 2 vectors, find the Area of that
  face ½ PQ x PR
• ½ i(1) j(0) k(0)
• ½ ½ lt1,0,0gt
• ½ v(12 02 02)
• ½

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Finding the Volume (cont.)

• Then, using those same 2 vectors and point
  P(1,1,1), find the equation of the PlanePQR.
• In the previous slide, the cross product of those
  2 vectors was determined lt1,0,0gt
• The general equation of a plane is
• A(x-x0) B(y-y0) C(z-z0) 0 where ltA,B,Cgt is
  lt1,0,0gt and ltx0,y0,z0gt is lt1,1,1gt
• The equation of PlanePQR is then1(x-1) 0

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Finding the Volume (cont.)

• The formula for the distance between a point and
  a plane is
• D Ax1 By1 Cz1 D / v(A2 B2 C2)
• Using the equation of the plane, x-1 0, and the
  point S(3,-1,2), the distance will be
• D 1(3) 0(-1) 0(2) (-1) / v(1 0 0)
• 2

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Finding the Volume (cont.)

• Now, going back to the volume of a tetrahedron
  formula
• V 1/3 d(s1, s2, s3), PlanePQR ½ PQ x PR
• Plug in all the components
• V 1/3(2)(1/2)
• 1/3

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Problem 3

• 3. Suppose the tetrahedron in the figure has a
  tri-rectangular vertex S. (This means that the 3
  angles at S are all right angles.) Let A, B, and
  C be the areas of the 3 faces that meet at S, and
  let D be the area of the opposite face PQR. Using
  the result of Problem 1, show that
• D2 A2 B2 C2

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3-D Pythagorean Theorem

• Going back to the results of Problem 1, the areas
  of the faces are
• A ½ qr
• B ½ pr
• C ½ pq
• D ½ v(-pr)2 (pq)2 (qr)2

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3-D Pythagorean Theorem (cont.)

• Therefore, D2 A2 B2 C2
• (½ v(-pr)2 (pq)2 (qr)2)2
• (½ qr)2 (½ pr)2 (½ pq)2
• ¼(pr)2 (pq)2 (qr)2 ¼ (qr)2 ¼ (pr)2
  (pq)2
• The 3-D Pythagorean Theorem is verified