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The three-dimensional counterpart of a triangle is a tetrahedron, which is a pyramid with four faces. We colour the four faces of such a tetrahedron blue, red, green and purple. Let B, R, G and P be the areas of these faces. In the diagram below, we've also drawn a blue, a red, a green and a purple vector. Let's call these vectors b, r, g, and p. The blue vector is perpendicular to the blue face, its length is equal to the area of the blue face, and the vector is pointing outside the tetrahedron. We have analogous statements for the other three vector/face combinations. (a) Show that b+r+ g+p = 0. Hint. Start by making one of the corners of the tetrahedron the origin 0 = (0,0, 0). Then the tetrahedron can be described by the vectors u = (u1, u2, 43), v = (v1,v2, v3) and w = (w1, w2, w3), as shown in the diagram below. Now express the vectors b, r, g, p in terms of u, v, w and go through the algebra.

The three-dimensional counterpart of a triangle is a tetrahedron, which is a pyramid with four faces. We colour the four faces of such a tetrahedron blue, red, green and purple. Let B, R, G and P be the areas of these faces. In the diagram below, we've also drawn a blue, a red, a green and a purple vector. Let's call these vectors b, r, g, and p. The blue vector is perpendicular to the blue face, its length is equal to the area of the blue face, and the vector is pointing outside the tetrahedron. We have analogous statements for the other three vector/face combinations. (a) Show that b+r+ g+p = 0. Hint. Start by making one of the corners of the tetrahedron the origin 0 = (0,0, 0). Then the tetrahedron can be described by the vectors u = (u1, u2, 43), v = (v1,v2, v3) and w = (w1, w2, w3), as shown in the diagram below. Now express the vectors b, r, g, p in terms of u, v, w and go through the algebra.

Elementary Geometry For College Students, 7e
Elementary Geometry For College Students, 7e
7th Edition
ISBN:9781337614085
Author:Alexander, Daniel C.; Koeberlein, Geralyn M.
Publisher:Alexander, Daniel C.; Koeberlein, Geralyn M.
ChapterP: Preliminary Concepts
SectionP.CT: Test
Problem 1CT
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The three-dimensional counterpart of a triangle is a tetrahedron, which is a pyramid with four faces. We colour the four faces of such a tetrahedron blue, red, green and purple. Let B, R, G and P be the areas of these faces. In the diagram below, we've also drawn a blue, a red, a green and a purple vector. Let's call these vectors b, r, g, and p. The blue vector is perpendicular to the blue face, its length is equal to the area of the blue face, and the vector is pointing outside the tetrahedron. We have analogous statements for the other three vector/face combinations. (a) Show that b +r+ g+p = 0. Hint. Start by making one of the corners of the tetrahedron the origin 0 = (0,0,0). Then the tetrahedron can be described by the vectors u = (u1, 42, u3), v = (v1,v2, v3) and w = (wi, w2, w3), as shown in the diagram below. Now express the vectors b, r, g, p in terms of u, v, w and go through the algebra. u
Transcribed Image Text:The three-dimensional counterpart of a triangle is a tetrahedron, which is a pyramid with four faces. We colour the four faces of such a tetrahedron blue, red, green and purple. Let B, R, G and P be the areas of these faces. In the diagram below, we've also drawn a blue, a red, a green and a purple vector. Let's call these vectors b, r, g, and p. The blue vector is perpendicular to the blue face, its length is equal to the area of the blue face, and the vector is pointing outside the tetrahedron. We have analogous statements for the other three vector/face combinations. (a) Show that b +r+ g+p = 0. Hint. Start by making one of the corners of the tetrahedron the origin 0 = (0,0,0). Then the tetrahedron can be described by the vectors u = (u1, 42, u3), v = (v1,v2, v3) and w = (wi, w2, w3), as shown in the diagram below. Now express the vectors b, r, g, p in terms of u, v, w and go through the algebra. u
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