Tetrahidral Project

# Tetrahidral Project - TheGeometryofa Tetrahedron Footnote...

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The Geometry of a  Tetrahedron Footnote 18:Section 10.4 Mark Jeng Professor Brewer

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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.
Problem 1 1. Let v 1 , v 2 , v 3 , and v 4 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: v 1 + v 2 + v 3 + v 4 = 0

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Setting up the vectors The area of a triangle is: ½|a x b| |v 1 | = ½ |QS x SR| |v 2 | = ½ |PS x PR| |v 3 | = ½ |PS x PQ| |v 4 | = ½ |PQ x PR|
Vector 1 = ½ |i(0) – j(0) + k(-qr)| |v 1 | = ½ = ½ |<0,0,-qr>| = ½ qr The direction of the vector is pointing outward from the xy-axis, so therefore the final vector is: v 1 = <0,0,-½ qr>

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Vector 2 = ½ |i(pr) – j(0) + k(0)| |v 2 | = ½ = ½ |<pr,0,0>| = ½ pr The direction of the vector is pointing outward from the yz-axis, so therefore the final vector is: v 2 = <-½ pr ,0,0>
Vector 3 = ½ |i(0) – j(pq) + k(0)| |v 3 | = ½ = ½ |<0,-pq,0>| = ½ pq The direction of the vector is pointing outward from the xz-axis, so therefore the final vector is: v 3 = <0,-½ pq,0>

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